Inverses of Bipartite Graphs

Abstract

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].

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Correspondence to Dong Ye.

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Corresponding author partially supported by a grant from National Natural Sciences Foundation of China (No. 11671347).

Partially supported by a grant from Simons Foundation (No. 359516).

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Yang, Y., Ye, D. Inverses of Bipartite Graphs. Combinatorica 38, 1251–1263 (2018). https://doi.org/10.1007/s00493-016-3502-y

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Mathematics Subject Classification (2000)

  • 05C22
  • 05C50
  • 06A07