Resistance matrix of a weighted graph

Abstract

In contrast to the classical notion of distance as the length of a shortest path between two vertices, the concept of resistance distance, introduced by Klein and Randic, arises naturally from several different considerations and is more amenable, to mathematical treatment. For a connected graph with n vertices, the resistance matrix of the graph is defined to be the n × n matrix with its (i, j)-entry equal to the resistance distance between the i-th and the j-th vertices. We obtain a formula for the inverse and the determinant of the resistance matrix of a weighted graph, thereby generalizing some earlier work, including that of Graham, Pollack, Lova'sz, Xiao and Gutman.