On group inverses of M-matrices with uniform diagonal entries

Abstract

In two recent works the condition of the diagonal entries of the group inverse of a singular and irreducible M-matrix being uniform (constant) has arisen: in resistive electrical circuits and in the effect upon the Perron root of certain diagonal perturbation of a nonnegative matrix. In this paper we first show a negative result that the group inverse of the Laplacian matrix of an undirected weighted graph G on n vertices with a cutpoint cannot have uniform diagonal. This includes the case when G is a tree. We characterize, however, all weighted n-cycles G whose Laplacian has a group inverse with a uniform diagonal. Finally, we consider the mean first passage matrix M of an ergodic Markov chain with a doubly stochastic transition matrix T. We show that if the group inverse of I− T has a uniform diagonal, then the group inverse of the M-matrix ρ( M) I− M is again an M-matrix.