一类双圈图的加权Kirchhoff指标  

A class of bicyclic graphs with weighted Kirchhoff index

作　　者：李成敏 边红[1] 于海征[2] 杨雪 LI Chengmin;BIAN Hong;YU Haizheng;YANG Xue(School of Mathematical Sciences,Xinjiang Normal University,Urumqi 830017,China;College of Mathematics and System Sciences,Xinjiang University,Urumqi 830046,China)

机构地区：[1]新疆师范大学数学科学学院,新疆乌鲁木齐830017 [2]新疆大学数学与系统科学学院,新疆乌鲁木齐830046

出　　处：《厦门大学学报（自然科学版）》2024年第6期1073-1081,共9页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(12361072,11971406);2023西藏自治区自然科学基金面上项目(2023D01A36);2023西藏自治区自然科学基金青年项目(2023D01B48);2022新疆师范大学创新团队(XJNU2022);2024新疆师范大学研究生创新基金(XSY202401025)。

摘　　要：[目的]探究当图G是一个具有n+2个顶点的顶点赋权双圈图,且其中两个圈只有一个公共顶点时,图G的加权Kirchhoff指标公式.[方法]通过比较系数法和枚举法,首先对双圈图G中任意两个顶点v_(i)和v_(j)的分布情况进行分类,然后讨论在S(G)-v_(i)-v_(j)中不被n-匹配M所饱和的e-顶点的分布情况.[结果]经过对顶点v_(i)和v_(j)不同的3种分布情形进行详细分类讨论,本文得出的双圈图的加权Kirchhoff指标公式在每种情形下都成立,且这一结果将Li,Li和Yan等关于单圈图的相应结果推广到了具有两个圈且仅有一个公共顶点的双圈图上.[结论]针对两个圈只有一个公共顶点的双圈图的加权Kirchhoff指标公式可通过计算获得,并可通过其对应的顶点-边赋权剖分图S(G)^(ω*)及其相应子图的匹配权和来表示.[Objective] We aim to investigate the weighted Kirchhoff index formula for bicyclic graphs G with n+2 vertices and consists of two cycles with a single common vertex.[Methods] By means of comparative coefficient and enumeration,we initially categorize the distribution of any two vertices v_(i) and v_(j) in the bicyclic graph G,and then discuss the distribution of e-vertices that are not saturated by an n-matching M in S(G)-v_(i)-v_(j).[Results] Following a comprehensive classification discussion of three distinct distributional cases of vertices v_(i) and v_(j),we establish that the weighted Kirchhoff index formula for bicyclic graphs,as derived in this paper,remains valid in each case.Subsequently,this result generalizes corresponding results of Li,Li,and Yan.on unicyclic graphs to bicyclic graphs with two cycles and only one common vertex.[Conclusion] This study demonstrates that the weighted Kirchhoff index formula for a bicyclic graph consisting of two cycles with a single common vertex can be computationally obtained and expressed for the sum of matching weights of its associated vertex-edge weighted subdivision graph S(G)^(ω^(*)) and its corresponding subgraphs.

关 键 词：加权Kirchhoff指标 双圈图 剖分图 完美匹配 

分 类 号：O157.5[理学—数学]