Some Bounds for the Kirchhoff Index of Graphs

Abstract

The resistance distance between two vertices of a connected graph $G$ is defined as the effective resistance between them in the corresponding electrical network constructed from $G$ by replacing each edge of $G$ with a unit resistor. The Kirchhoff index of $G$ is the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.