Inverses of Bipartite Graphs

TL;DR

This paper characterizes bipartite graphs with unique perfect matchings whose adjacency matrix inverses are diagonally similar to non-negative matrices, addressing an open problem and revealing connections to M"obius functions of posets.

Contribution

It provides a complete characterization of bipartite graphs with unique perfect matchings whose inverse adjacency matrices are diagonally similar to non-negative matrices, solving an open problem.

Findings

Characterization of bipartite graphs with unique perfect matchings and specific inverse properties

Connection established between graph inverses and M"obius functions of posets

Resolution of an open problem posed by Godsil in 1985

Abstract

Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to M\"obius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39].