Inverses of Bipartite Graphs
Let G be a bipartite graph with adjacency matrix A. If G has a unique perfect matching, then A has an inverse A1 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].
Mathematics Subject Classification (2000)05C22 05C50 06A07
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