Mathematical Researches

fa نتایجی در باب گراف‌های کامل علامت‌دار Some results on the signed complete graphs جبر alg مقاله مستقل Original Manuscript فرض کنید <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAQCAIAAADVkMW0AAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAdJJREFUSEvdVjFygzAQFHmLSeHxC8gLgCZVWneijJ8CJTzBVRrEC8ILPC4s/kJOQgJJEEtyIDOJGqTx3d7tnbTnoO979E/X0xa8uqbIsqLptsC+h9lB2EwJC31bd9E8inK6LqYHmhIe9XAwaoGJB5ZhajIjmINztmMkxwCq7+hs9x1TQJAbizm6AKDd/Vvu86YxcNlHvsfEuamqr5FmL4jLtqhXRfKZcXu8ZWaVBiTIQcSlOfbgpfuKk2PVRYVdtKTJgqX1UjiIRXe7RG/pjr3yOi3LeOehL+CLDs/So/moon3o6t5eKXLhFpeLzfx8tyfa1ee2PYVBeLrsU7u5ljnzRVUiy5pUrEiu3JidCzcfPMOWXlt2I+GStOfaoc2qu/AVdYW7PfXQnhFv8fIr0RplPNul5yvtDS2ZHtu0c33QuoePxJlastZI0jRXPSh7gpl0khw+o4Ry0VGlQgNSFMleGmUG+E4dGziD5knKdvPDEIUPAApLDAZgKcf8jAu/HgNbbdBZ4muz25brA79T6AnO78wxwYlPBTEqcvzzPzMMTxkz7L39+gJqorOQCaHOs9wz0Y11clHQYG4dXmOE4FslxxvyEna7RI4WAdTCw/xPmX4BmzYFNL3gC+cAAAAASUVORK5CYII=" style="width:54.75pt; height:12pt" >  یک گراف کامل علامتدار  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAAQCAIAAACk6KkqAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAHpJREFUKFO9kLsRgDAIhsFZPCsnSEawd4RkHVzD3riBE3gWkl0QiKW1FNzH434eKCLwZd1nVpM/FfYcEfMO1QHjUn2hQsRMIaREhUUJUtETBMyVBIHYyFoa2br1PsM89UbbCo3sDo2OcfCILyVYso1xoSbrCK8q/vCrB2aLTMDWIzBQAAAAAElFTkSuQmCC" style="width:6pt; height:12pt" >  رأسی باشد که یال‌های منفی آن زیر گراف  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAIlJREFUKFNj/P//PwNuwIRHDihFlPT2dEYQsJ54B6jjzkRrMC99O5ADtBsIbk+wYrCacBvCAfPStoFYUOltaUiy/4E8iOx/iN13bl1h0FFTgTpz+4ZZVlqqEA7MMBQfwO0B67597RjCYpDRCJOAunFZDLEbt8VAg4FawQDiVCgPbhUj7cMcZ7QBAB3hh1Q9e2mHAAAAAElFTkSuQmCC" style="width:7.5pt; height:12pt" >  را القا می‌کنند. اگر  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAIlJREFUKFNj/P//PwNuwIRHDihFlPT2dEYQsJ54B6jjzkRrMC99O5ADtBsIbk+wYrCacBvCAfPStoFYUOltaUiy/4E8iOx/iN13bl1h0FFTgTpz+4ZZVlqqEA7MMBQfwO0B67597RjCYpDRCJOAunFZDLEbt8VAg4FawQDiVCgPbhUj7cMcZ7QBAB3hh1Q9e2mHAAAAAElFTkSuQmCC" style="width:7.5pt; height:12pt" >   اجتماع مجزای دو گراف <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAQCAIAAACp9tltAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAALZJREFUKFOdUrERwyAMFMxiXOQ8AUyQzlPIS6TMANB6ClewiKvALkRgLnYSOyH5ipMe6f+BxRihDryOllh/UYMbFCtQJrxvK1ODUeIKF0/Ko9eya5sdYaUJUmfeBwD1LAKg/UKkoKqZkUO4zSBPImsjydnY4PYyTC4261cx3qLErQEOTdvBOLmcTjDTCNif6eS86Lv5aXhyQ/dzTaLVuBqk8nZqSuAIL9QfHvZoarK34CGC1X/CO+UOG2Tm6vZfAAAAAElFTkSuQmCC" style="width:10.5pt; height:12pt" >  و  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAQCAIAAACp9tltAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAALBJREFUKFOdksENwyAMRU1maXqoOgHsQ5bosQM4yziL9FTYhdqWSxI1tKj/ggQP8/8XrpQCfRr6MKH+QvMyBWcKc/58zabmOYx3uCV2XhL66/l0YMwOwaNyXwR8RhEg0g+Qi+omywD5+QB/GdUbW9Zg03LUoaTYPP82kyh6wf3qrBqw8GabQY3JNysrsXhDJ8gIjLuAhGsxijaUiLYFNtGEKP3ZIsMaqKQzVUOu/xO+AEzwUeq34DpyAAAAAElFTkSuQmCC" style="width:10.5pt; height:12pt" >  باشد، در این صورت <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAAQCAIAAAA0+9DlAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAZBJREFUSEvNVTtygzAUFDkLTpHxCeAEhCaV23SoTeNTuBGlfQRXaZBuwAk8KSzuQp5+IGFGfppknKgxst9K+5Z962wcR/Iv19Mtq0G0lLZi+AO+A9xM3c2gmb8kKwomF18+dOsYkCgv3mjhNFdAGBUbjmLqYycwCmu4BcxuBVN0nIb6ueFoQX2saWzmZWk7wyxek671mYVgLQycYFGSNQmsQqzdoQTT1yqJVibAc/5wvRS7OlfG7OrjscoTpgKwZPvsEOLzVLxsEuB9VLPJWtMbRTnMtR3ySBksdXFUM/nVq/Ogrj93iSlisbYVcMWsH0q5IjYBs8nmJ6xoIQJ2eJPZaQk0y+sd+TgI25NnlGrPph8ELbOyhTQuM/h0SgqaUQcE/I9MNnRnwvZhnpmp0O25udYbYzgdGRKWjRLeTN5ZzwSjUxBs9zVfT1rjXQ4BwSK5ZRnpHDE3AeJX/jjUkS6blprd7wk0cKrCIVyikxdxtF8Sz7O1KQILbd8qbaXT6/uVpGQcaiptUQY0U+ofV/sNHmXnI9nLw8QAAAAASUVORK5CYII=" style="width:38.25pt; height:12pt" >  با نماد <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:59.25pt; height:12pt" >  نشان داده می‌شود. فرض کنید  <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:21pt; height:14.25pt" >  ماتریس مجاورت <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAQCAIAAABV4/KnAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAE9JREFUKFNj/P//PwMGYMIUAopgF2X4f3uCFZr6tG0MQHNB4mnbgAwQ2JYGZLOgm+s5878nDnNJcQN+tVdu3UG2F+FeuOP+/2ekOBywuwEA8wsqLwcQ0MEAAAAASUVORK5CYII=" style="width:5.25pt; height:12pt" > باشد. چندجمله‌ای ویژه و مقادیر ویژه  <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:21pt; height:14.25pt" > ، به ترتیب، چندجمله‌ای ویژه و مقادیر ویژه <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAQCAIAAABV4/KnAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAE9JREFUKFNj/P//PwMGYMIUAopgF2X4f3uCFZr6tG0MQHNB4mnbgAwQ2JYGZLOgm+s5878nDnNJcQN+tVdu3UG2F+FeuOP+/2ekOBywuwEA8wsqLwcQ0MEAAAAASUVORK5CYII=" style="width:5.25pt; height:12pt" > نامیده می‌شوند. بزرگترین مقدار ویژه  <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:21pt; height:14.25pt" > نیز اندیس گراف <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAQCAIAAABV4/KnAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAE9JREFUKFNj/P//PwMGYMIUAopgF2X4f3uCFZr6tG0MQHNB4mnbgAwQ2JYGZLOgm+s5878nDnNJcQN+tVdu3UG2F+FeuOP+/2ekOBywuwEA8wsqLwcQ0MEAAAAASUVORK5CYII=" style="width:5.25pt; height:12pt" > نامیده می‌شود. در این مقاله، به مطالعه چندجمله‌ای ویژه گراف علامت‌دار <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:64.5pt; height:12pt" >  می‌پردازیم که در آن  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAQCAIAAACZeshMAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAANlJREFUOE+1kcENgzAMRZ3u0ktHCPuEdcIavTdswARVD012Se0fBwgqUkHqvxDHfMd+NjlnOqvLWaP4/mIeeyPqhkSUhg6B6cdNnzwzKzhyoRxV0VuyPiLC2YUSrERIsrf+WHPLVfTui1H+I5StWirwrQRi1Oe3z8K823SpuO1oXUNop/fT3q4Ni/iaysN2uj+Y2Y6AY3/ghgazwYgMDw1w20qaM16BC4YKf3VmK+afV8NmJTbvgnMQ7CXZ7il4ZQhgRxTDsu9j5ugxmX7Kqn5UnaeOlI1AO6sPRrj4vT2FTwwAAAAASUVORK5CYII=" style="width:15pt; height:12pt" >   اجتماع مجزای <m:r> </m:r> <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAQCAIAAABLKsIUAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAHZJREFUKFNj/P//PwMOwIRLAihOLbnt6Yzp26H2oJp5Z/uGWVZaqjA3AN0JBbcnWMEdZjXhNtD9CDkga1saQ9o2uAiKmXduXUEyEcUPd7auYgjzVoEbjKzv9rVjOmoqDHe2p0+EOBXJPqhjrNK2AR0CBIx0Dk8A1WdTHnppSSMAAAAASUVORK5CYII=" style="width:6.75pt; height:12pt" >  گراف کامل  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAQCAIAAACp9tltAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAALNJREFUKFOVkMsRwyAMREXKwv3gduQ2co/cgUvIRe6F6AMD8RAP2Yul8WO1Usg5w5wec5hSf6P7GlTLdgKc22JNWPfLRMkqYowQkVudyJtO4DWlSjKmAaaMo2KqnooV66tlRXW8q4YYkNkuwO/DTePxfMlmP/QVtEXWVJRsWExkQySrji9dXwtoueW5/wWpTNZ6aLHpLkVYNi3HGq1hp6H26g5lRB9ln3LXkWlNVsPloPicPjFyZThI1kuQAAAAAElFTkSuQmCC" style="width:10.5pt; height:12pt" > ، و  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAIlJREFUKFNj/P//PwNuwIRHDihFlPT2dEYQsJ54B6jjzkRrMC99O5ADtBsIbk+wYrCacBvCAfPStoFYUOltaUiy/4E8iOx/iN13bl1h0FFTgTpz+4ZZVlqqEA7MMBQfwO0B67597RjCYpDRCJOAunFZDLEbt8VAg4FawQDiVCgPbhUj7cMcZ7QBAB3hh1Q9e2mHAAAAAElFTkSuQmCC" style="width:7.5pt; height:12pt" >  یک زیرگراف دلخواه  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAQCAIAAABGNLJTAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAANFJREFUKFOdkrERwyAMRUVmSVJ5AjxBUnmDdHgd3HqEVGkCG3gCLoVhFyIhHJ8vOMflN6C7h/T1bRFjhGodqkkC/6FtL0jtEADC0KZC9PZ7LvpGeS1Bar/eleFiK+DSqAX2WpVJwpjG1tSZyDyg0HihyQdrcVOGY8rEvyZuLaf7Exfd18b0an/XCflQZs0m342imIzG42MQsF1SYngBSaF4VI4V3+Xdc4LFwZlKmTLw48vbh2vOR8oAupMNtP0+HWbXdBf8E2Y3Xm8z4DsQlHm13krjQNZ0H/YeAAAAAElFTkSuQmCC" style="width:11.25pt; height:12pt" >    است. همچنین، برای یک خانواده از کاکتوس‌های  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAIlJREFUKFNj/P//PwNuwIRHDihFlPT2dEYQsJ54B6jjzkRrMC99O5ADtBsIbk+wYrCacBvCAfPStoFYUOltaUiy/4E8iOx/iN13bl1h0FFTgTpz+4ZZVlqqEA7MMBQfwO0B67597RjCYpDRCJOAunFZDLEbt8VAg4FawQDiVCgPbhUj7cMcZ7QBAB3hh1Q9e2mHAAAAAElFTkSuQmCC" style="width:7.5pt; height:12pt" > ، چندجمله‌ای ویژه گراف علامت‌دار  <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:54.75pt; height:12pt" >  را محاسبه نموده و یک کران دقیق برای اندیس این گراف ارائه می‌دهیم.    <div style="text-align: justify;">Let <m:omath><m:r> </m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:57pt; height:12pt" > be a signed complete graph of order <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAAQCAIAAACk6KkqAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAHpJREFUKFO9kLsRgDAIhsFZPCsnSEawd4RkHVzD3riBE3gWkl0QiKW1FNzH434eKCLwZd1nVpM/FfYcEfMO1QHjUn2hQsRMIaREhUUJUtETBMyVBIHYyFoa2br1PsM89UbbCo3sDo2OcfCILyVYso1xoSbrCK8q/vCrB2aLTMDWIzBQAAAAAElFTkSuQmCC" style="width:6pt; height:12pt" >  whose negative edges induce a subgraph <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAIlJREFUKFNj/P//PwNuwIRHDihFlPT2dEYQsJ54B6jjzkRrMC99O5ADtBsIbk+wYrCacBvCAfPStoFYUOltaUiy/4E8iOx/iN13bl1h0FFTgTpz+4ZZVlqqEA7MMBQfwO0B67597RjCYpDRCJOAunFZDLEbt8VAg4FawQDiVCgPbhUj7cMcZ7QBAB3hh1Q9e2mHAAAAAElFTkSuQmCC" style="width:7.5pt; height:12pt" >. If  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAIlJREFUKFNj/P//PwNuwIRHDihFlPT2dEYQsJ54B6jjzkRrMC99O5ADtBsIbk+wYrCacBvCAfPStoFYUOltaUiy/4E8iOx/iN13bl1h0FFTgTpz+4ZZVlqqEA7MMBQfwO0B67597RjCYpDRCJOAunFZDLEbt8VAg4FawQDiVCgPbhUj7cMcZ7QBAB3hh1Q9e2mHAAAAAElFTkSuQmCC" style="width:7.5pt; height:12pt" >  is a disjoint :union: of  two graphs <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAQCAIAAACp9tltAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAALZJREFUKFOdUrERwyAMFMxiXOQ8AUyQzlPIS6TMANB6ClewiKvALkRgLnYSOyH5ipMe6f+BxRihDryOllh/UYMbFCtQJrxvK1ODUeIKF0/Ko9eya5sdYaUJUmfeBwD1LAKg/UKkoKqZkUO4zSBPImsjydnY4PYyTC4261cx3qLErQEOTdvBOLmcTjDTCNif6eS86Lv5aXhyQ/dzTaLVuBqk8nZqSuAIL9QfHvZoarK34CGC1X/CO+UOG2Tm6vZfAAAAAElFTkSuQmCC" style="width:10.5pt; height:12pt" >  and <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAQCAIAAACp9tltAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAALBJREFUKFOdksENwyAMRU1maXqoOgHsQ5bosQM4yziL9FTYhdqWSxI1tKj/ggQP8/8XrpQCfRr6MKH+QvMyBWcKc/58zabmOYx3uCV2XhL66/l0YMwOwaNyXwR8RhEg0g+Qi+omywD5+QB/GdUbW9Zg03LUoaTYPP82kyh6wf3qrBqw8GabQY3JNysrsXhDJ8gIjLuAhGsxijaUiLYFNtGEKP3ZIsMaqKQzVUOu/xO+AEzwUeq34DpyAAAAAElFTkSuQmCC" style="width:10.5pt; height:12pt" > , then we denote <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:38.25pt; height:12pt" >  by <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:59.25pt; height:12pt" > . Let <m:omath><m:r> </m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAATCAIAAADnDkEVAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAStJREFUOE+1lDEWgjAMhqtnAQYfJ+gNdHJydesKB3B0Y6GjV3ByKieQE/AYaO+iKQKmEJSHkonX5v+SR/6GPaaEVqkQqaJTtRJwqalLNgGuBBeKVHdinXJOFPhOV4KJka5xZyT/Gx1Ek+BQiGijR4cMB4bg9nMQ+HcQjayxwMhb4SJ0mfONX+d40d225/JCJPeCkBWVwUBMz5Jyf3IyTFWwMPCGPb9OvOgSoUt/w/NS0/RMVvvLFjLGWDPO296NrIJoWwN69WdAO8mLbuQxjncrCD/Of8H1tDU9S66H9rVY13RBDOpDbWSBJgvoRp7ZqZ2OnSMa/HBQ43TKAvDMWWezxtPIdW8TI79Tj37KayK2zpKbwJZbcotZ/twNvALtHy1IOXIxvLPF/l7lCQcHNlodT88rAAAAAElFTkSuQmCC" style="width:23.25pt; height:14.25pt" > be the adjacency matrix of  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAIAAABvtHqPAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAEZJREFUKFNj/P//PwM2wIRVFChIRQmGCVaYtlhNuP2fKf/o/21pDCAmDNyeoIPdcpX8mfkq1HQV6R5E8geyk///Z6Re6AIAh3IoKvPTcO0AAAAASUVORK5CYII=" style="width:6pt; height:12.75pt" > . The characteristic polynomial and eigenvalues of<m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAATCAIAAADnDkEVAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAATRJREFUOE+1lDFygzAQRUXOAhQeTqAbOJWrtOnUwgFcunODSq6QypU4QTgBQ4F0F3slIrREYizsYStG2v92h/1act8UUtSM1SKskYLBpUSXZAtcMMoEVvtiWVPqCmygC0bYSte4CuZH00EUBYdCro01OmQsYAiuP71Av+Pucj/8RDhR/NYvEXLo6CEzyWn5q9tb8grESfOC9KOCkyC9vQ6ns80wMjX2pMjTYCumYFOiy+xAu0GG6S0fT80RMtZY8ede74qPeXk0gKn+O/GPrvh3VX0mEFnVvcOdtEt6e/35sq9Fu2YON6iYkrMFMF3xCznb6eg5ToM3MQ8qgo4sYJ8ZLAkYo7XZn6eR65yJkd+xK917ff6aAutnz02gy+25xTR/4wZOQBIxqBdTwnvmRZgn25f+AC9aNjYRUzyuAAAAAElFTkSuQmCC" style="width:23.25pt; height:14.25pt" >  are called the characteristic polynomial and eigenvalues of <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAARCAIAAABvtHqPAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAEZJREFUKFNj/P//PwM2wIRVFChIRQmGCVaYtlhNuP2fKf/o/21pDCAmDNyeoIPdcpX8mfkq1HQV6R5E8geyk///Z6Re6AIAh3IoKvPTcO0AAAAASUVORK5CYII=" style="width:6pt; height:12.75pt" > , respectively. The index of <m:omath><m:r> </m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAFJJREFUKFNj/P//PwNuwIRHDihFQJrh/+0JVmgmpG0D2ggBDEAMUgAX2paGYP//j2G458z/Mz3hxlHmNOrovnLrDtYAQvgbybtwfzPSMswJeAwASbNFq4D03bIAAAAASUVORK5CYII=" style="width:7.5pt; height:12pt" > is the largest eigenvalue of  <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:21pt; height:14.25pt" > . In this paper, we study the characteristic polynomial of <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFkAAAAQCAIAAADyPkRcAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAitJREFUWEftVrF1wjAQtTMLTsHLBGaCQJ8RTAmjyCUegSoNYgI8AY8CeRdyJ50syZZsGSjyEtRgQKf79/3179Lb7Za8lmTgreWhOZbrdXls/j4xDRS69hUKuoAlWJ4zoZ7/xfIVnASY4IXUhyQI4pRYCh7Fkx3bBkfGRiVwN90Jtc8GcuEVBRKgpSKfCx6tGztWUWmYIOT6JjpqhP9cygJy7ewLQh3K1cWFvun5TdIO5xBOwYoJPLix9C1SFHFcCANNCeReqF2mjXe6ntlcz/nXaoY2c1jtdp+zCZYKscnHu444flf5PJsQPri1KRfZskrqbZami1Ia/SNQ64sw6QJcNId9jfmy7Xm+msIDYoPYpFqmtJYVkvosLmabU6ue0waPfQSqiyrAhbjUeENARfX+MLHNUiwZHCA3Ghln5Hwdy4YysHX2CNTEVazXO80N7N7NcZ93I3oWMHSA67J+J+te8gBUMBXZ/MDyQxm72dA7+zntXdYzHp8zzjCL8X+PqTtdY9Lcgs5PRi04tvJeKZQOamVYZRiqHJcGXoVFKu2SXKi2qtPqTiS/q9lCNlQBixotsKInM3/HVGc5g8a4ohQSeqE6azeMhh3V4kNQ2yDOAhOkLS9dAnEhQUDvZMNDBHEguyz1M1b82nlVcH81CN+aEqBuVYLhIuK9ARWSA45HcRE9e0Uc/fQtgtEdkh9RawoXoCVzbybZQBSUZ26yJs7IOQ+yp6iN15IM/AAGdF7bv9MHxgAAAABJRU5ErkJggg==" style="width:66.75pt; height:12pt" >  where <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAQCAIAAACZeshMAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAANlJREFUOE+1kcENgzAMRZ3u0ktHCPuEdcIavTdswARVD012Se0fBwgqUkHqvxDHfMd+NjlnOqvLWaP4/mIeeyPqhkSUhg6B6cdNnzwzKzhyoRxV0VuyPiLC2YUSrERIsrf+WHPLVfTui1H+I5StWirwrQRi1Oe3z8K823SpuO1oXUNop/fT3q4Ni/iaysN2uj+Y2Y6AY3/ghgazwYgMDw1w20qaM16BC4YKf3VmK+afV8NmJTbvgnMQ7CXZ7il4ZQhgRxTDsu9j5ugxmX7Kqn5UnaeOlI1AO6sPRrj4vT2FTwwAAAAASUVORK5CYII=" style="width:15pt; height:12pt" >  is the disjoint :union: of <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAYAAAAQCAIAAAC6IZmZAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAHBJREFUKFNj/P//PwMqYELjA7kkCW1PZ0zfDjUDrPHO9g2zrLRUYcb+vz3BCm6F1YTbQAcAHfF/WxpD2jYQAwxAGu/cuoKkDeSIO1tXMYR5q8B1A4VuXzumo6YCtCN9IsTS/zDzrdK2Ac0GAkYq+hEAYKc4OZbu2i8AAAAASUVORK5CYII=" style="width:4.5pt; height:12pt" >  complete graphs <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAQCAIAAACp9tltAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAALNJREFUKFOVkMsRwyAMREXKwv3gduQ2co/cgUvIRe6F6AMD8RAP2Yul8WO1Usg5w5wec5hSf6P7GlTLdgKc22JNWPfLRMkqYowQkVudyJtO4DWlSjKmAaaMo2KqnooV66tlRXW8q4YYkNkuwO/DTePxfMlmP/QVtEXWVJRsWExkQySrji9dXwtoueW5/wWpTNZ6aLHpLkVYNi3HGq1hp6H26g5lRB9ln3LXkWlNVsPloPicPjFyZThI1kuQAAAAAElFTkSuQmCC" style="width:10.5pt; height:12pt" >  and <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAQCAIAAACgHXkXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAIlJREFUKFNj/P//PwNuwIRHDihFlPT2dEYQsJ54B6jjzkRrMC99O5ADtBsIbk+wYrCacBvCAfPStoFYUOltaUiy/4E8iOx/iN13bl1h0FFTgTpz+4ZZVlqqEA7MMBQfwO0B67597RjCYpDRCJOAunFZDLEbt8VAg4FawQDiVCgPbhUj7cMcZ7QBAB3hh1Q9e2mHAAAAAElFTkSuQmCC" style="width:7.5pt; height:12pt" >  is an arbitrary subgraph of  <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAQCAIAAACUZLgLAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAANhJREFUOE+lU8sRwiAQXaxFPaUCUoGe0oE30g65WoInL0IHqSDjIdAL7vKRiTMBo+/CwuzbffsA5pyD7dhtpxDjL5ruGaEdLIAdWr9hvS4owdkQRnLg0uRYqLBZAYRzJRLLSFGhUH6gYTPqRZTYstQq0UhiQBJaYTnvpHmOoRkfbw905QssBssj1kWSRKGyozFWgsxVEpesHev6XKCA4JPDkJysNIh4K1ggGfWmFdTEdH8lH2mFx6XvU3Pck2HQHbRdWLVOs/PUdCd8bPN0PV9mwAIZ7LeP8wIwsYEKK+QDpQAAAABJRU5ErkJggg==" style="width:13.5pt; height:12pt" >  Also, we compute the characteristic polynomial of <img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:61.5pt; height:12pt" > for some classes of cactus graphs <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAQCAIAAABCwWJuAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAJ1JREFUKFNj/P//PwMRgIkINSAlpKnbns4IAtYT7wC13ploDealb0e2C+g+ILg9wYrBasJtCAfMS9sG5UAoBgi1LQ1JGYiHpuw/xH13bl1h0FFTgdqzfcMsKy1VNA/C7EERRjgBZjnYvNvXjiHJAG1FmA3XTozjgGqA5uF3HDDIwAEEtAUMIB6E8jCDCBouKGGFxNk2IQ0SqowDlA4Adk3YHsgkiXMAAAAASUVORK5CYII=" style="width:9.75pt; height:12pt" >  and provide a sharp bound for the index of <img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAQCAIAAABV4/KnAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAE9JREFUKFNj/P//PwMGYMIUAopgF2X4f3uCFZr6tG0MQHNB4mnbgAwQ2JYGZLOgm+s5878nDnNJcQN+tVdu3UG2F+FeuOP+/2ekOBywuwEA8wsqLwcQ0MEAAAAASUVORK5CYII=" style="width:5.25pt; height:12pt" >.</div>   اندیس, چندجمله‌ای ویژه, گراف علامت‌دار, گراف کامل, کاکتوس Index, Characteristic polynomial, Signed graph, Complete graph, Cactus 18 32 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1693-3&slc_lang=fa&sid=1 Farideh Heydari فریده حیدری f-heydari@kiau.ac.ir 10031947532846006661 10031947532846006661 Yes Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran گروه ریاضی، واحد کرج، دانشگاه آزاد اسلامی، کرج، ایران