{
  "id": "0911.4741",
  "version": "v1",
  "published": "2009-11-24T23:52:18.000Z",
  "updated": "2009-11-24T23:52:18.000Z",
  "title": "The spectrum of random k-lifts of large graphs (with possibly large k)",
  "authors": [
    "Roberto Imbuzeiro Oliveira"
  ],
  "journal": "Journal of Combinatorics 1 (3/4) p.285-306, 2011",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We study random k-lifts of large, but otherwise arbitrary graphs G. We prove that, with high probability, all eigenvalues of the adjacency matrix of the lift that are not eigenvalues of G are of the order (D ln (kn))^{1/2}, where D is the maximum degree of G. Similarly, and also with high probability, the \"new\" eigenvalues of the Laplacian of the lift are all in an interval of length (ln (nk)/d)^{1/2} around 1, where d is the minimum degree of G. We also prove that, from the point of view of Spectral Graph Theory, there is very little difference between a random k_1k_2 ... k_r-lift of a graph and a random k_1-lift of a random k_2-lift of ... of a random k_r-lift of the same graph. The main proof tool is a concentration inequality for sums of random matrices that was recently introduced by the author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-24T23:52:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large graphs",
      "possibly large",
      "high probability",
      "eigenvalues",
      "spectral graph theory"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.4741I"
    }
  }
}