【作者】 张倩；

【导师】 王燕；

【作者基本信息】 烟台大学 ， 数学， 2025， 硕士

【摘要】 电阻距离是图的一个内在度量,已经被许多科研工作者广泛研究.一般用连通图来模拟电网络,设Γ是连通图,图Γ的每条边看作单位电阻,定义Γ的任意两点间的电阻距离为该电网络中两点之间的等效电阻.本文利用群表示论和线性代数的相关知识,计算了凯莱图的电阻距离和基尔霍夫指标.文章先计算得到群G=D_2n×Z₂的凯莱图的特征值及其相应的标准正交特征向量,其中D_2n=〈a,b|aⁿ=(b²=1,b^-1ab=a^-1〉是一个2n)阶的二面体群,Z₂=〈c〉是一个2阶循环群,进而计算出该凯莱图任意两点间的电阻距离和基尔霍夫指标.类似地,推广得到群G=D_2n×Z_m的凯莱图任意两点间的电阻距离和基尔霍夫指标.本文的研究内容共分为三章.具体内容如下.第一章主要介绍了本文所用到的符号、研究背景、国内外研究现状等,并综述了计算凯莱图的电阻距离和基尔霍夫指标所用到的引理.第二章中,详细计算了Cay（G,S）的特征值和相应的标准正交的特征向量,其中G=D_2n×Z₂,S=S₁∪bS₂∪cS₃∪cbS₄,S_i,1≤i≤4,S是D_2n中循环子群的子集,进而计算得出任意两点间的电阻距离和基尔霍夫指标.第三章中,利用类似的方法将第二章结论进行了推广,求出了Cay（G,S）的特征值和相应的标准正交的特征向量,其中G=D_2n×Z_m,S=S₁∪bS₂∪C₃S₃∪C₄bS₄.S_i,1≤i≤4是D_2n中循环子群的子集,C_j,j=3,4是Z_m的子集.从理论上来说,可以得到图中任意两点间的电阻距离和基尔霍夫指标.更多还原

【Abstract】 The resistance distance is an intrinsic metric of a graph,which has been extensively studied by many researchers.In general,a connected graph is used to model an electrical,and each edge of the graphΓis regarded as a unit resistor.The resistance distance between any two points of the graphΓis defined as the equivalent between the two points in this electrical network.In this paper,by using relevant knowledge of group representation theory and linear algebra,we calculate the resistance distance and the Kirchhoff index of Cayley graphs.The eigenvalues and their corresponding orthonormal eigenvectors of the Cayley graph of the groupG=D_2n×Z₂are calculated,whereD_2n=〈a,b|aⁿ=(b²=1,b^-1ab=a^-1〉is a dihedral group of order 2n,and Z₂=〈c〉is a cyclic group of order 2.The resistance distance and the Kirchhoff index between any two points of the Cayley graph are then calculated.Similarly,the resistance distance and Kirchhoff index between any two points of the Cayley graph of the group G=D_2n×Z_m are extended.The research content of this paper is divided into three chapters.The specific content is as follows.Chapter 1 mainly introduces the symbols used in this paper,the research background,the current state of research both domestically and internationally,and summarizes the lemmas used to calculate the resistance distance and Kirchhoff index of the Cayley graph.In Chapter 2,the eigenvalues and corresponding orthonormal eigenvectors of Cay（G,S）are detailedly calculated,where G=D_2n×Z₂,S=S₁∪bS₂∪cS₃∪cbS₄,S_i,1≤i≤4,S,is a subset of cyclic subgroups of D_2n,and then the resistance distance and Kirchhoff index between any two points are calculated.In Chapter 3,a similar method is used to generalize the conclusions of Chapter 2,and the eigenvalues of Cay（G,S） and the corresponding standard orthogonal eigenvectors are obtained,where G=D_2n×Z_m,S=S₁∪bS₂∪C₃S₃∪C₄bS₄.S_i,1≤i≤4 is a subset of the cyclic subgroups in D_2n,C_j,j=3,4 is a subset of Z_m.Theoretically,the resistance distance between any two points in the graph and the Kirchhoff indicator can be obtained.更多还原