fa مقادیر ویژه ی ماتریس فاصله ی گراف های کیلی جبر alg علمی پژوهشی کاربردی S در این مقاله، ماتریس فاصله و چند جمله ای مشخصه ی یک گراف کیلی روی گروه متناهی <em><span dir="LTR">G</span></em> بر حسب نمایش های تحویل ناپذیر گروه <em><span dir="LTR">G</span></em> بیان می شوند. فرمول های دقیقی برای مقادیر ویژه ی ماتریس فاصله ی گراف های کیلی مکعبی روی گروه های آبلی و برخی گراف های شناخته شده ی دیگر ارائه می دهیم. خانواده ی نامتناهی از گراف های کیلی که تمام مقادیر ویژه ی ماتریس فاصله ی آن ها اعداد صحیحی هستند، معرفی می کنیم. ثابت می کنیم روی گروه آبلی متناهی <em><span dir="LTR">G</span></em> یک گراف کیلی مکعبی همبند وجود دارد که تمام مقادیر ویژه ی ماتریس فاصله ی آن صحیح هستند اگر و تنها اگر <em><span dir="LTR">G</span></em> یکریخت با یکی از گروه های Z_4&nbsp;، Z_6&nbsp;، Z_4xZ_2&nbsp;، Z_6xZ_2&nbsp;&nbsp;یا Z_2xZ_2xZ_2&nbsp;&nbsp;باشد. علاوه بر این نشان می دهیم که، تحت یکریختی، تنها 5 گراف کیلی مکعبی همبند وجود دارد که تمام مقادیر ویژه ی ماتریس فاصله ی آن ها صحیح هستند. <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">In this paper, graphs are undirected and loop-free and groups are finite. By </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>C</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>n</i></m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;we mean the cycle graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices, the complete graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices and the complete bipartite graph with parts size </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, respectively. Also by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, we mean the cyclic group of order </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and the symmetric group on </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;symbols, respectively.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a simple connected graph with vertex set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance</i> between vertices </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is the length of a shortest path between them. The <i>distance matrix</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is an </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r><m:r>&times;</m:r><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;matrix whose </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>i</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>j</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-entry is </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance characteristic polynomial</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&chi;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>det(</m:r><m:r><i>&lambda;I</i></m:r><m:r><i>-</i></m:r><m:r><i>D</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and its zeros are the <i>distance eigenvalues</i> (in short </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues) of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. If </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalue of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with multiplicity </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, then we denote it by </span></span><m:omath><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>[</m:r><m:r><i>m</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>]</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Let </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;&hellip;&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance spectral radius</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and we denote it by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&rho;</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Also the multiset </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>S</m:r><m:r><i>pe</i></m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>c</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2].</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a group and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a subset of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;not containing the identity element of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>Cayley graph</i> of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with respect of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is a graph with vertex set </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and edge set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{{</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>sg</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}|</m:r><m:r><i>g</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>s</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a simple </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-regular graph. Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>xg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>yg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent. This implies that </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1,</m:r><m:r><i>h</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r></span></span><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="subSup"><m:suphide m:val="on"><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:suphide></m:limloc></m:chr></m:narypr><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:sub><m:sup></m:sup><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:">&zwj;</span></span></m:e></m:nary><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance (adjacency) integral</i> if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;=C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a Cayley graph over a finite group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. It is well-known that one can determine the (adjacency) eigenvalues </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;by the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;in terms of the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prims, hexagonal torus network and cubic Cayley graphs over abelian groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Results and discussion</span></b></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;admits a connected cubic distance integral Cayley graph if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is isomorphic to one of the groups </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, or </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Furthermore, up to isomorphism, there are exactly </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>5</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;connected cubic distance integral Cayley graphs over Abelian groups which are </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3,3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prism.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Conclusion</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The following conclusions were drawn from this research.</span></span></span></span> <ul> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">The characteristic polynomial of the distance matrix of Cayley graphs over a group <i>G</i> is determined by the irreducible representations of <i>G</i>.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Exact formulas for </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups</span></span><span style="font-size:10.0pt"> are given.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Infinite family of distance integral Cayley graphs are constructed.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.<span new="" roman="" style="font-family:" times="">In this paper, graphs are undirected and loop-free and groups are finite. By </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>C</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>n</i></m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;we mean the cycle graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices, the complete graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices and the complete bipartite graph with parts size </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, respectively. Also by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, we mean the cyclic group of order </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and the symmetric group on </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;symbols, respectively.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a simple connected graph with vertex set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance</i> between vertices </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is the length of a shortest path between them. The <i>distance matrix</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is an </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r><m:r>&times;</m:r><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;matrix whose </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>i</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>j</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-entry is </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance characteristic polynomial</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&chi;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>det(</m:r><m:r><i>&lambda;I</i></m:r><m:r><i>-</i></m:r><m:r><i>D</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and its zeros are the <i>distance eigenvalues</i> (in short </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues) of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. If </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalue of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with multiplicity </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, then we denote it by </span></span><m:omath><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>[</m:r><m:r><i>m</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>]</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Let </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;&hellip;&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance spectral radius</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and we denote it by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&rho;</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Also the multiset </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>S</m:r><m:r><i>pe</i></m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>c</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2].</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a group and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a subset of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;not containing the identity element of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>Cayley graph</i> of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with respect of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is a graph with vertex set </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and edge set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{{</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>sg</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}|</m:r><m:r><i>g</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>s</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a simple </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-regular graph. Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>xg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>yg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent. This implies that </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1,</m:r><m:r><i>h</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r></span></span><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="subSup"><m:suphide m:val="on"><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:suphide></m:limloc></m:chr></m:narypr><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:sub><m:sup></m:sup><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:">&zwj;</span></span></m:e></m:nary><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance (adjacency) integral</i> if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;=C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a Cayley graph over a finite group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. It is well-known that one can determine the (adjacency) eigenvalues </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;by the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;in terms of the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prims, hexagonal torus network and cubic Cayley graphs over abelian groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Results and discussion</span></b></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;admits a connected cubic distance integral Cayley graph if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is isomorphic to one of the groups </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, or </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Furthermore, up to isomorphism, there are exactly </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>5</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;connected cubic distance integral Cayley graphs over Abelian groups which are </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3,3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prism.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Conclusion</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The following conclusions were drawn from this research.</span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">In this paper, graphs are undirected and loop-free and groups are finite. By </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>C</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>n</i></m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;we mean the cycle graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices, the complete graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices and the complete bipartite graph with parts size </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, respectively. Also by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, we mean the cyclic group of order </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and the symmetric group on </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;symbols, respectively.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a simple connected graph with vertex set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance</i> between vertices </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is the length of a shortest path between them. The <i>distance matrix</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is an </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r><m:r>&times;</m:r><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;matrix whose </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>i</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>j</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-entry is </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance characteristic polynomial</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&chi;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>det(</m:r><m:r><i>&lambda;I</i></m:r><m:r><i>-</i></m:r><m:r><i>D</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and its zeros are the <i>distance eigenvalues</i> (in short </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues) of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. If </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalue of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with multiplicity </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, then we denote it by </span></span><m:omath><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>[</m:r><m:r><i>m</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>]</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Let </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;&hellip;&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance spectral radius</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and we denote it by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&rho;</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Also the multiset </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>S</m:r><m:r><i>pe</i></m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>c</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2].</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a group and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a subset of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;not containing the identity element of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>Cayley graph</i> of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with respect of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is a graph with vertex set </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and edge set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{{</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>sg</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}|</m:r><m:r><i>g</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>s</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a simple </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-regular graph. Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>xg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>yg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent. This implies that </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1,</m:r><m:r><i>h</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r></span></span><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="subSup"><m:suphide m:val="on"><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:suphide></m:limloc></m:chr></m:narypr><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:sub><m:sup></m:sup><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:">&zwj;</span></span></m:e></m:nary><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance (adjacency) integral</i> if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;=C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a Cayley graph over a finite group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. It is well-known that one can determine the (adjacency) eigenvalues </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;by the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;in terms of the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prims, hexagonal torus network and cubic Cayley graphs over abelian groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Results and discussion</span></b></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;admits a connected cubic distance integral Cayley graph if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is isomorphic to one of the groups </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, or </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Furthermore, up to isomorphism, there are exactly </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>5</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;connected cubic distance integral Cayley graphs over Abelian groups which are </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3,3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prism.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Conclusion</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The following conclusions were drawn from this research.</span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">The characteristic polynomial of the distance matrix of Cayley graphs over a group <i>G</i> is determined by the irreducible representations of <i>G</i>.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Exact formulas for </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups</span></span><span style="font-size:10.0pt"> are given.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Infinite family of distance integral Cayley graphs are constructed.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.<span new="" roman="" style="font-family:" times="">In this paper, graphs are undirected and loop-free and groups are finite. By </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>C</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>n</i></m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;we mean the cycle graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices, the complete graph with </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;vertices and the complete bipartite graph with parts size </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, respectively. Also by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, we mean the cyclic group of order </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and the symmetric group on </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;symbols, respectively.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a simple connected graph with vertex set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance</i> between vertices </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is the length of a shortest path between them. The <i>distance matrix</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is an </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r><m:r>&times;</m:r><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;matrix whose </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>i</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>j</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-entry is </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>i</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>v</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>j</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>distance characteristic polynomial</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&chi;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>det(</m:r><m:r><i>&lambda;I</i></m:r><m:r><i>-</i></m:r><m:r><i>D</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and its zeros are the <i>distance eigenvalues</i> (in short </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues) of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. If </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalue of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with multiplicity </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>m</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, then we denote it by </span></span><m:omath><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>[</m:r><m:r><i>m</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>]</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Let </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&ge;&hellip;&ge;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance spectral radius</i> of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and we denote it by </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&rho;</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Also the multiset </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>&hellip;</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&lambda;</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>S</m:r><m:r><i>pe</i></m:r></span></span><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>c</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>D</m:r></span></span></i></m:sub></m:ssub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(&Gamma;)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2].</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a group and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a subset of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;not containing the identity element of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. The <i>Cayley graph</i> of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;with respect of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>S</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, denoted by </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, is a graph with vertex set </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and edge set </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>{{</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>sg</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}|</m:r><m:r><i>g</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>s</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>}</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is a simple </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>|</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-regular graph. Let </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><i>&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>xg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>yg</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are adjacent. This implies that </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1,</m:r><m:r><i>h</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>-</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:sup></m:ssup><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>g</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r><m:r><i>d</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(1)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;for all </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>g</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>h&isin;</i></m:r><m:r><i>G</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)=</m:r></span></span><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="subSup"><m:suphide m:val="on"><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:suphide></m:limloc></m:chr></m:narypr><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>y</m:r><m:r>&isin;</m:r><m:r>G</m:r></span></span></i></m:sub><m:sup></m:sup><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:">&zwj;</span></span></m:e></m:nary><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is called <i>distance (adjacency) integral</i> if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;Let </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;=C</m:r><m:r><i>ay</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>G</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r><i>S</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;be a Cayley graph over a finite group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. It is well-known that one can determine the (adjacency) eigenvalues </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;by the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&Gamma;</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;in terms of the irreducible representations of </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prims, hexagonal torus network and cubic Cayley graphs over abelian groups.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Results and discussion</span></b></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="tab-stops:center 240.0pt right 475.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;admits a connected cubic distance integral Cayley graph if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>G</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is isomorphic to one of the groups </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, or </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>&times;</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>Z</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">. Furthermore, up to isomorphism, there are exactly </span></span><m:omath><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>5</m:r></span></span></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;connected cubic distance integral Cayley graphs over Abelian groups which are </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>K</m:r></span></span></i></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3,3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>3</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>4</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;and </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>6</m:r></span></span></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">, where </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:ctrlpr></m:ctrlpr></span></span></m:ssubpr><m:e><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>P</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">&nbsp;is the </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prism.</span></span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Conclusion</span></b></span></span></span><br> <span style="font-size:11pt"><span style="line-height:12.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The following conclusions were drawn from this research.</span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">The characteristic polynomial of the distance matrix of Cayley graphs over a group <i>G</i> is determined by the irreducible representations of <i>G</i>.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Exact formulas for </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups</span></span><span style="font-size:10.0pt"> are given.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Infinite family of distance integral Cayley graphs are constructed.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The characteristic polynomial of the distance matrix of Cayley graphs over a group <i>G</i> is determined by the irreducible representations of <i>G</i>.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">Exact formulas for </span></span><m:omath><i><span style="font-size:10.0pt"><span cambria="" math="" style="font-family:"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span new="" roman="" style="font-family:" times="">-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups</span></span><span style="font-size:10.0pt"> are given.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Infinite family of distance integral Cayley graphs are constructed.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups.</span></span></span></span></span></li> <li style="margin-top:8px"><span style="font-size:11pt"><span style="line-height:12.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt">One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.</span></span></span></span></span></li> </ul> ماتریس فاصله, گراف کیلی, مقدار ویژه, نمایش تحویل ناپذیر Distance matrix, rreducible representation, Cayley graph, Eigenvalue. 1 18 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1255-1&slc_lang=fa&sid=1 Majid Arezoomand مجید آرزومند arezoomand@lar.ac.ir 10031947532846005455 10031947532846005455 Yes University of Larestan مجتمع آموزش عالی لارستان