{
  "id": "1904.07766",
  "version": "v1",
  "published": "2019-04-16T15:38:22.000Z",
  "updated": "2019-04-16T15:38:22.000Z",
  "title": "Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs",
  "authors": [
    "Jun Ge",
    "Fengming Dong"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph $G(m,n,p)=K_{m,n}-pK_2$ $(p\\leq \\min\\{m,n\\})$, which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in $G(n,n,p)$. As a corollary, we obtain the Kirchhoff index of $G(m,n,p)$ which extends a previous result by Shi and Chen.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-16T15:38:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "05C05"
    ],
    "keywords": [
      "spanning trees",
      "resistance distance",
      "effective resistance",
      "complete bipartite graph containing",
      "simple formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}