Mathematical Researches

fa مجتمع های ساده گون تجزیه پذیر رأسی نظیر به گراف های مسیر Vertex Decomposable Simplicial Complexes Associated to Path Graphs جبر alg مقاله مستقل Original Manuscript شناخت مجتمعهای سادهگون تجزیهپذیر رأسی به واسطۀ خواص جبری و توپولوژیکیای که دارند از جمله مسائل مهم در جبر جابه‌جایی ترکیبیاتی به‌شمار میرود. در این راستا معرفی خانوادههایی از مجتمعهای سادهگون با این خاصیت بسیار مورد توجه است. در این مقاله مجتمع سادهگون استنلی-ریزنر نظیر به ایدهآل t-خوشهای گرافهای مکمل مسیر بررسی شده است. برای این خانواده از مجتمعهای سادهگون، مجموعۀ رویههای آن‌ها را به‌طور دقیق مشخص کرده و با استفاده از این موضوع نشان میدهیم این دسته از مجتمعهای سادهگون دارای خاصیت تجزیهپذیری رأسی هستند. در واقع با توجه به محض بودن آن ها ثابت میشود که حلقۀ استنلی-ریزنر آن‌ها دارای خاصیت کوهن-مکالی است. از آن‌جاکه 2-خوشه ایدهآل‌ها همان ایدهآلهای یالی گرافها هستند، این دسته از مجتمعهای سادهگون شامل خانوادۀ مجتمعهای سادهگون مستقلهای گراف مکمل مسیر هستند. در پایان به‌عنوان نتیجه نشان میدهیم که ایدهآل <img alt="" height="18" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="6" >-مستقل‌های گراف مکمل مسیر یک ایدهآل جداشوندۀ رأسی است و جداساز بتی آن را ارائه میدهیم. Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="39" >. Later Bjorner and Wachs extended this concept to non-pure complexes. Being defined in an inductive way, vertex decomposable simplicial complexes are considered as a well behaved class of complexes and has been studied in many research papers. Because of their interesting algebraic and topological properties, giving a characterization for this class of complexes is of great importance and is one of the main problems in combinatorial commutative algebra. In this regard obtaining families of simplicial complexes with this property is of great interest. In this paper we present a new family of vertex decomposable simplicial complexes, which is associated to the t-clique ideal of the complement of path graphs. The t-clique ideal is a natural generalization of the concept of the edge ideal of a graph. For a graph G, a complete subgraph of G with t vertices is called a t-clique of G. The ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="37" > generated by the monomials <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="52" > of degree t such that the induced subgraph of G on the set <img alt="" height="22" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="70" > is a complete graph, is called the t-clique ideal of G. We consider the Stanley- Reisner simplicial complex of the ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="43" >, where <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="14" > is a path graph of order n.  For such a simplicial complex <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif" width="9" >, we obtain the set of facets of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif" width="9" > and using this characterization we show that every such simplicial complex is vertex decomposable, whose shedding vertex is an endpoint of the path graph. Indeed, any simplicial complex in this family is Cohen-Macaulay, since it is pure. Since edge ideals of graphs are in fact 2-clique ideals, this family of simplicial complexes contains the independence complexes of complement of path graphs. Finally, as a consequence it is shown that the t-independence ideal of the complement of a path graph is vertex splittable and its Betti splitting is presented Material and methods To prove the vertex decomposability of <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="42" >, first we characterize the set of facets of  <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="42" >. This helps us to find a shedding vertex for this simplicial complex and then by an inductive approach the vertex decomposability has been proved. Results and discussion For positive integers <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif" width="9" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="6" >, we show that a subset F of the vertex set of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="14" > is a facet of <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="42" > if and only if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="78" > and every component of the induced subgraph <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="34" > is a path graph of even order. Using this characterization, it is shown that any endpoint of the path graph is a shedding vertex of <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="42" > and  <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="42" > is vertex decomposable. Moreover, it is proved that the ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif" width="66" > has a Betti splitting. Conclusion The following conclusions were drawn from this research. <ul> <li>A characterization for the set of facets of the simplicial complex <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="42" > is presented.</li> <li>The simplicial complex <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="42" > is vertex decomposable for any positive integers <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif" width="9" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="6" >.</li> <li>The ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif" width="66" > has a Betti splitting for any any positive integers <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif" width="9" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="6" >.<a href="./files/site1/files/51/%D9%85%D8%B1%D8%A7%D8%AF%DB%8C.pdf">./files/site1/files/51/%D9%85%D8%B1%D8%A7%D8%AF%DB%8C.pdf</a></li> </ul> ایده‌آل t-خوشه ای, تجزیه پذیری رأسی, گراف مسیر, مجتمع ساده گون. T-clique ideal, Vertex decomposability, Path graph, Simplicial complex. 79 84 http://mmr.khu.ac.ir/browse.php?a_code=A-10-132-1&slc_lang=fa&sid=1 Somayeh Moradi سمیه مرادی somayeh.moradi1@gmail.com 10031947532846005733 10031947532846005733 Yes Ilam University دانشگاه ایلام، دانشکدۀ علوم، گروه ریاضی