Inverses of Unipathic M-Matrices

Abstract

In this paper we characterize all nonnegative matrices whose inverses are M-matrices with unipathic digraphs. A digraph is called unipathic if there is at most one simple path from any vertex j to any other vertex k. The set of unipathic digraphs on n vertices includes the simple n-cycle and all digraphs whose underlying undirected graphs are trees (or forests). Our results facilitate the construction of nonnegative matrices whose inverses are M-matrices with unipathic digraphs. We highlight this procedure for inverses of tridiagonal M-matrices and of M-matrices whose digraphs are simple n-cycles with loops.