广义四元数群的幂图的电阻距离  

Resistance distance of the power graph of generalized quaternion group

作　　者：潘向峰[1] 徐思奥 李云翔 PAN Xiangfeng;XU Si’ao;LI Yunxiang(School of Mathematical Sciences,Anhui University,Hefei 230601,Anhui)

机构地区：[1]安徽大学数学科学学院,安徽合肥230601

出　　处：《长江大学学报（自然科学版）》2022年第2期105-112,共8页Journal of Yangtze University(Natural Science Edition)

基　　金：安徽省自然科学基金项目“图的控制及相关问题研究”(2108085MA02);安徽省高校自然科学基金项目“基于电阻距离的图结构分析”(KJ2020A0001)。

摘　　要：图G中任意两点u和v之间的电阻距离R_(G)(u,v)等于把G中每条边都替换为单位电阻后得到的电网络中对应两点间的有效电阻。一个有限群Ω的幂图,记为P(Ω),它的顶点集是Ω且其中任意两个不同元素是相邻的当且仅当其中一个元素是另一个元素的整数幂。若一个有限群Q_(4n)可以表示为〈x,y|x^(2n)=l,x^(n)=y^(2),yxy^(-1)=x^(-1)〉,其中l是单位元,且n=2^(k),k∈N+,则称Q_(4n)为广义四元数群。利用电网络理论中串并联原理、星三角变换、消去原理和星网变换,得到了广义四元数群Q_(4n)的幂图P(Q_(4n))中任意两点之间的电阻距离,并进一步得到了P(Q_(4n))的电阻直径、基尔霍夫指数、度积基尔霍夫指数、度和基尔霍夫指数。The resistance distance R_(G)(u,v)between two vertices u and v of a graph G is equal to the effectiveresistance between the two vertices in the corresponding electrical network in which each edge of G is replaced byaunit resistor.The power graph of a finite groupΩ,denoted by P(Ω),is the graph with vertex setΩ,and any two different vertices are adjacent if and only if one is a power of the other.If a finite group Q_(4n) can be expressed as〈x,y|x^(2n)=l,xn=y^(2),yxy^(-1)=x^(-1)〉,where l is the identity element and n=2k,k∈N+,then it is called a generalized quaternion group.In this paper,the resistance distance between any two vertices was obtained in the power graph P(Q_(4n))of the generalized quaternion group Q_(4n) using some principles and techniques in electrical network theory,i.e.,series and parallel principles,star-triangle transformation,the principle of elimination,and the star-mesh transformation,and then the resistance diameter,the Kirchhoff index,multiplicative degree-Kirchhoff index、additive degree-Kirchhoffindex of P(Q_(4n))were derived.

关 键 词：电阻距离 电阻直径 基尔霍夫指数 度积基尔霍夫指数 度和基尔霍夫指数 

分 类 号：O157.5[理学—数学]