图运算下的基尔霍夫指数

Kirchhoff Indices under Graph Operations

设G是一个无向的连通图,电阻距离Ω_(G)(u,v)是指将图G中的每条边用单位电阻代替后,G中两点u和v在对应电网络中的有效电阻,G中所有无序顶点对的电阻距离之和是图G的基尔霍夫指数。记RK_(a)(G)是将G的每条边变换为a阶的完全图K_(a)得到的图,利用电网络原理给出图RK_(a)(G)各点对之间的电阻距离、基尔霍夫指数、度和与度积基尔霍夫指数,得到这些指数与原图G的相应基尔霍夫指数之间的关系。

Let G be an undirected connected graph,The resistance distanceΩ_(G)(u,v)between two vertices u,v of G is equal to the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor.The sum of the resistance distances of all unordered vertex pairs in the graph is the Kirchhoff index of G.Let RK_(a)(G)be the graph obtained by turning each edge of G into a K_(a)(the complete graph of ordera).The resistance distance,Kirchhoff index,additive degree-Kirchhoff index and multiplicative degree-Kirchhoff index of the graph RK_(a)(G)are studied by electrical network theory,and the relationship between these indices and the corresponding Kirchhoff indices of the original graph G is obtained.