Resistance Distances and Kirchhoff Indices Under Graph Operations

Yujun Yang,Yue Yu

doi:10.1109/access.2020.2995935

Abstract

The resistance distance between any two vertices of a connected graph G is defined as the net effective resistance between them in the electrical network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is defined as the sum of resistance distances between all pairs of vertices. In this paper, two unary graph operations on G are taken into consideration, with the resulted graphs being denoted by RT(G) and H(G). Using electrical network approach and combinatorial approach, we derive explicit formulae for resistance distances and Kirchhoff indices of RT(G) and H(G). It turns out that resistance distances and Kirchhoff indices of RT(G) and H(G) could be expressed in terms of resistance distances and graph invariants of G. Our result generalizes the previously known result on the Kirchhoff index of RT(G) for a regular graph G to the Kirchhoff index of RT(G) for an arbitrary graph G.

Highlights

Let G be a connected graph with vertex set V (G) and edge set E(G)
It is well known that distance functions are of fundamental to a graph
The most natural and best known distance function defined on a graph is the distance, where the distance between any two vertices of G is defined as the length of a shortest path connecting them

Summary

INTRODUCTION

Let G be a connected graph with vertex set V (G) and edge set E(G). Suppose that V (G) = {v1, v2, . . . , vn}. Resistance distances and Kirchhoff indices of Q-double join graphs [21] and other two novel graph operations were obtained [22]. In [29], Liu et al studied a new graph operation RT (G) and obtained Kirchhoff index of RT (G) for a regular graph G. Resistance distances and Kirchhoff indices of stellated graphs were obtained Motivated by these results, in this paper, we take two unary graph operations into consideration. Resistance distances and the Kirchhoff index of RT (G) for a general graph G are determined, which generalized the result obtained by Liu et al in [29].

RESISTANCE DISTANCES AND THE

CONCLUSION