Inverses of Bipartite Graphs

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 Möbius 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].

9 pages, 2 figures