{
  "id": "1201.5909",
  "version": "v2",
  "published": "2012-01-27T23:03:46.000Z",
  "updated": "2012-06-01T18:35:39.000Z",
  "title": "Brownian approximation to counting graphs",
  "authors": [
    "Soumik Pal"
  ],
  "comment": "9 pages; a previous error corrected in the main result, Introduction expanded",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to infinity is known. It has been observed that, if k_n=o(\\sqrt{n}), this limit is asymptotically equal to the $k_n$th moment of the area under the standard Brownian excursion. These moments have been computed in the literature via independent methods. In this article we show why this is true for k_n=o(\\sqrt[3]{n}) starting from an observation made by Joel Spencer. The elementary argument uses a result about strong embedding of the Uniform empirical process in the Brownian bridge proved by Komlos, Major, and Tusnady.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-01T18:35:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "05C30"
    ],
    "keywords": [
      "brownian approximation",
      "counting graphs",
      "standard brownian excursion",
      "brownian bridge",
      "uniform empirical process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.5909P"
    }
  }
}