{
  "id": "0712.3035",
  "version": "v2",
  "published": "2007-12-18T20:00:21.000Z",
  "updated": "2009-08-09T19:08:12.000Z",
  "title": "Identities and Inequalities for Tree Entropy",
  "authors": [
    "Russell Lyons"
  ],
  "comment": "12 pages; revision contains more background",
  "journal": "Combin. Probab. Comput. 19, no. 2 (2010), 303-313.",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede-Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lueck's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-08-09T19:08:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "60C05",
      "05C80"
    ],
    "keywords": [
      "tree entropy respects stochastic domination",
      "identities",
      "inequalities",
      "establishes luecks determinant conjecture",
      "random infinite rooted graphs"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.3035L"
    }
  }
}