{
  "id": "1903.05462",
  "version": "v1",
  "published": "2019-03-13T12:55:37.000Z",
  "updated": "2019-03-13T12:55:37.000Z",
  "title": "On the location of roots of the independence polynomial of bounded degree graphs",
  "authors": [
    "Pjotr Buys"
  ],
  "comment": "11 pages, 1 figure",
  "categories": [
    "math.CO",
    "math-ph",
    "math.DS",
    "math.MP"
  ],
  "abstract": "In [1] Peters and Regts confirmed a conjecture by Sokal by showing that for every $\\Delta \\in \\mathbb{Z}_{\\geq 3}$ there exists a complex neighborhood of the interval $\\left[0, \\frac{\\left(\\Delta - 1\\right)^{\\Delta - 1}}{\\left(\\Delta-2\\right)^\\Delta}\\right)$ on which the independence polynomial is nonzero for all graphs of maximum degree $\\Delta$. Furthermore, they gave an explicit neighborhood $U_\\Delta$ containing this interval on which the independence polynomial is nonzero for all finite rooted Cayley trees with branching number $\\Delta$. The question remained whether $U_\\Delta$ would be zero-free for the independence polynomial of all graphs of maximum degree $\\Delta$. In this paper it is shown that this is not the case. [1] Han Peters and Guus Regts, On a conjecture of sokal concerning roots of the independence polynomial, Michigan Math. J. (2019), Advance publication.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-13T12:55:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C31",
      "37F10"
    ],
    "keywords": [
      "independence polynomial",
      "bounded degree graphs",
      "maximum degree",
      "finite rooted cayley trees",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}