{
  "id": "1003.0356",
  "version": "v3",
  "published": "2010-03-01T14:20:35.000Z",
  "updated": "2011-12-02T14:57:06.000Z",
  "title": "The number of graphs and a random graph with a given degree sequence",
  "authors": [
    "Alexander Barvinok",
    "J. A. Hartigan"
  ],
  "comment": "52 pages, minor improvements",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0-1 matrices with prescribed row and column sums.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-12-02T14:57:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "05C07",
      "05C30",
      "52B55",
      "60F05"
    ],
    "keywords": [
      "random graph",
      "efficient asymptotic formula approximating",
      "tame degree sequence",
      "maximum entropy matrix",
      "prescribed degrees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.0356B"
    }
  }
}