{
  "id": "1205.3529",
  "version": "v1",
  "published": "2012-05-15T23:45:48.000Z",
  "updated": "2012-05-15T23:45:48.000Z",
  "title": "The entropy of random-free graphons and properties",
  "authors": [
    "Hamed Hatami",
    "Serguei Norine"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Every graphon defines a random graph on any given number $n$ of vertices. It was known that the graphon is random-free if and only if the entropy of this random graph is subquadratic. We prove that for random-free graphons, this entropy can grow as fast as any subquadratic function. However, if the graphon belongs to the closure of a random-free graph property, then the entropy is $O(n \\log n)$. We also give a simple construction of a non-stepfunction random-free graphon for which this entropy is linear, refuting a conjecture of Janson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-15T23:45:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random graph",
      "random-free graph property",
      "non-stepfunction random-free graphon",
      "graphon defines",
      "subquadratic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.3529H"
    }
  }
}