{
  "id": "1711.00175",
  "version": "v1",
  "published": "2017-11-01T02:48:12.000Z",
  "updated": "2017-11-01T02:48:12.000Z",
  "title": "The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic",
  "authors": [
    "Alexander Mednykh",
    "Ilya Mednykh"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we develop a new method to produce explicit formulas for the number $\\tau(n)$ of spanning trees in the undirected circulant graphs $C_{n}(s_1,s_2,\\ldots,s_k)$ and $C_{2n}(s_1,s_2,\\ldots,s_k,n).$ Also, we prove that in both cases the number of spanning trees can be represented in the form $\\tau(n)=p \\,n \\,a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending only of parity of $n.$ Finally, we find an asymptotic formula for $\\tau(n)$ through the Mahler measure of the associated Laurent polynomial $L(z)=2k-\\sum\\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-01T02:48:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "39A10"
    ],
    "keywords": [
      "spanning trees",
      "arithmetic properties",
      "produce explicit formulas",
      "integer sequence",
      "prescribed natural number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}