Group inverses of matrices with path graphs

Abstract

A simple formula for the group inverse of a 2 × 2 block matrix with a bipartite digraph is given in tenus of the block matrices. This formula is used to give a graph-theoretic description of the group inverse of an irreducible tridiagonal matrix of odd order with zero diagonal (which is singular). Relations between the zero/nonzero structures of the group inverse and the Moore-Penrose inverse of such matrices are given. An extension of the graph-theoretic description of the group inverse to singular matrices with tree graphs is conjectured.