{
  "id": "0909.3354",
  "version": "v1",
  "published": "2009-09-18T04:43:20.000Z",
  "updated": "2009-09-18T04:43:20.000Z",
  "title": "The Number of Independent Sets in a Regular Graph",
  "authors": [
    "Yufei Zhao"
  ],
  "comment": "5 pages. Accepted by Combin. Probab. Comput",
  "journal": "Combinatorics, Probability and Computing, 19 (2010), No. 2, 315-320",
  "doi": "10.1017/S0963548309990538",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that the number of independent sets in an N-vertex, d-regular graph is at most (2^{d+1} - 1)^{N/2d}, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also works for a weighted generalization, i.e., an upper bound for the independence polynomial of a regular graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-18T04:43:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "independent sets",
      "complete d-regular bipartite graphs",
      "bipartite case",
      "disjoint union",
      "general case"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.3354Z"
    }
  }
}