{
  "id": "math/0012033",
  "version": "v1",
  "published": "2000-12-05T22:00:11.000Z",
  "updated": "2000-12-05T22:00:11.000Z",
  "title": "On the chromatic roots of generalized theta graphs",
  "authors": [
    "Jason Brown",
    "Carl Hickman",
    "Alan D. Sokal",
    "David G. Wagner"
  ],
  "comment": "LaTex2e, 25 pages including 2 figures",
  "journal": "J. Combin. Theory B 83, 272-297 (2001)",
  "categories": [
    "math.CO",
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The generalized theta graph \\Theta_{s_1,...,s_k} consists of a pair of endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \\ge 1. We prove that the roots of the chromatic polynomial $pi(\\Theta_{s_1,...,s_k},z) of a k-ary generalized theta graph all lie in the disc |z-1| \\le [1 + o(1)] k/\\log k, uniformly in the path lengths s_i. Moreover, we prove that \\Theta_{2,...,2} \\simeq K_{2,k} indeed has a chromatic root of modulus [1 + o(1)] k/\\log k. Finally, for k \\le 8 we prove that the generalized theta graph with a chromatic root that maximizes |z-1| is the one with all path lengths equal to 2; we conjecture that this holds for all k.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-12-05T22:00:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "30C15",
      "82B20"
    ],
    "keywords": [
      "chromatic root",
      "k-ary generalized theta graph",
      "path lengths equal",
      "internally disjoint paths",
      "chromatic polynomial"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....12033B"
    }
  }
}