Inverses of Unipathic M-Matrices

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.