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arXiv · 1611.06535

Inverses of Bipartite Graphs

Yujun Yang·Dong Ye
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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].

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Yujun Yang, Dong Ye. 2016-11-20. Inverses of Bipartite Graphs. https://doi.org/10.1007/s00493-016-3502-y

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