{
  "id": "1902.08493",
  "version": "v1",
  "published": "2019-02-22T14:03:00.000Z",
  "updated": "2019-02-22T14:03:00.000Z",
  "title": "A general bridge theorem for self-avoiding walks",
  "authors": [
    "Christian Lindorfer"
  ],
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $X$ be an infinite, locally finite, connected, quasi-transitive graph without loops or multiple edges. A graph height function on $X$ is a map adapted to the graph structure, assigning to every vertex an integer, called height. Bridges are self-avoiding walks such that heights of interior vertices are bounded by the heights of the start- and end-vertex. The number of self-avoiding walks and the number of bridges of length $n$ starting at a vertex $o$ of $X$ grow exponentially in $n$ and the bases of these growth rates are called connective constant and bridge constant, respectively. We show that for any graph height function $h$ the connective constant of the graph is equal to the maximum of the two bridge constants given by increasing and decreasing bridges with respect to $h$. As a concrete example, we apply this result to calculate the connective constant of the Grandparent graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-22T14:03:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "82B20"
    ],
    "keywords": [
      "general bridge theorem",
      "self-avoiding walks",
      "graph height function",
      "connective constant",
      "bridge constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}