{
  "id": "1204.0463",
  "version": "v2",
  "published": "2012-04-02T16:38:15.000Z",
  "updated": "2015-02-19T18:27:40.000Z",
  "title": "Benjamini--Schramm continuity of root moments of graph polynomials",
  "authors": [
    "Péter Csikvári",
    "Péter E. Frenkel"
  ],
  "comment": "22 pages. Minor corrections made",
  "categories": [
    "math.CO"
  ],
  "abstract": "Recently, M.\\ Ab\\'ert and T.\\ Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Ab\\'ert and Hubai proved that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized log of the chromatic polynomial converges to a harmonic real function outside a bounded disc. In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number $v_0$ the measures arising from the Tutte polynomial $Z_{G_n}(z,v_0)$ converge in holomorphic moments if the sequence $(G_n)$ of finite graphs is Benjamini--Schramm convergent. This answers a question of Ab\\'ert and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-04-02T16:38:15.000Z",
      "comment": "22 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-19T18:27:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C31",
      "05C15",
      "05C40",
      "05C60"
    ],
    "keywords": [
      "graph polynomials",
      "root moments",
      "benjamini-schramm continuity",
      "benjamini-schramm convergent",
      "finite graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.0463C"
    }
  }
}