{
  "id": "2311.05542",
  "version": "v1",
  "published": "2023-11-09T17:44:26.000Z",
  "updated": "2023-11-09T17:44:26.000Z",
  "title": "Counterexamples to conjectures on the occupancy fraction of graphs",
  "authors": [
    "Stijn Cambie",
    "Jorik Jooken"
  ],
  "comment": "8 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The occupancy fraction of a graph is a (normalized) measure on the size of independent sets under the hard-core model, depending on a variable (fugacity) $\\lambda.$ We present a criterion for finding the graph with minimum occupancy fraction among graphs with a fixed order, and disprove five conjectures on the extremes of the occupancy fraction and (normalized) independence polynomial for certain graph classes of regular graphs with a given girth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-09T17:44:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C31",
      "05C35",
      "05C69",
      "68R05",
      "68R10"
    ],
    "keywords": [
      "conjectures",
      "counterexamples",
      "minimum occupancy fraction",
      "independent sets",
      "regular graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}