{
  "id": "1611.06535",
  "version": "v1",
  "published": "2016-11-20T15:55:54.000Z",
  "updated": "2016-11-20T15:55:54.000Z",
  "title": "Inverses of Bipartite Graphs",
  "authors": [
    "Yujun Yang",
    "Dong Ye"
  ],
  "comment": "9 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "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].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-20T15:55:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bipartite graph",
      "unique perfect matching",
      "adjacency matrix",
      "symmetric integral matrix",
      "strong connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}