On resistance matrices of weighted balanced digraphs

Abstract

Let G be a connected graph with . Then the resistance distance between any two vertices i and j is given by , where is the th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix , there is an elegant formula to compute the inverse of R. This says that where A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.