{
  "id": "1812.04484",
  "version": "v1",
  "published": "2018-12-10T02:34:00.000Z",
  "updated": "2018-12-10T02:34:00.000Z",
  "title": "Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics",
  "authors": [
    "Alexander Mednykh",
    "Ilya Mednykh"
  ],
  "comment": "17 pages. arXiv admin note: text overlap with arXiv:1711.00175",
  "categories": [
    "math.CO"
  ],
  "abstract": "In the present paper, we investigate a family of circulant graphs with non-fixed jumps $$G_n=C_{\\beta n}(s_1, \\ldots,s_k,\\alpha_1n,\\ldots,\\alpha_\\ell n),\\, 1\\le s_1<\\ldots<s_k\\le[\\frac{\\beta n}{2}],\\, 1\\le \\alpha_1< \\ldots<\\alpha_\\ell\\le[\\frac{\\beta}{2}].$$ Here $n$ is an arbitrary large natural number and integers $s_1, \\ldots,s_k,\\alpha_1, \\ldots,\\alpha_\\ell$ are supposed to be fixed. First, we present an explicit formula for the number of spanning trees in the graph $G_n.$ This formula is a product of $\\beta s_k-1$ factors, each given by the $n$-th Chebyshev polynomial of the first kind evaluated at the roots of some prescribed polynomial of degree $s_k.$ Next, we provide some arithmetic properties of the complexity function. We show that the number of spanning trees in $G_n$ 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 of parity of $\\beta$ and $n.$ Finally, we find an asymptotic formula for $\\tau(n)$ through the Mahler measure of the Laurent polynomials differing by a constant from $2k-\\sum\\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-10T02:34:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "39A10"
    ],
    "keywords": [
      "circulant graphs",
      "arithmetic properties",
      "non-fixed jumps",
      "asymptotic",
      "arbitrary large natural number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}