{
  "id": "1807.06465",
  "version": "v1",
  "published": "2018-07-16T01:04:30.000Z",
  "updated": "2018-07-16T01:04:30.000Z",
  "title": "Invertibility of adjacency matrices for random $d$-regular graphs",
  "authors": [
    "Jiaoyang Huang"
  ],
  "comment": "This is the first draft. Suggestions are welcome. arXiv admin note: text overlap with arXiv:1806.01382",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Very recently, M\\'esz\\'ados [22], and Nguyen and Wood [24] proved that the adjacency matrices of random $d$-regular graphs obtained from the union of $d$ random perfect matchings are nonsingular with high probability. This answers an open problem by Frieze [12] and Vu [29,30] for random $d$-regular graphs with even number of vertices. In this paper, we study random $d$-regular graphs obtained from the configuration model, and prove that with high probability their adjacency matrices are nonsingular. The proof combines a local central limit theorem and a large deviation estimate, which generalizes the argument developed for studying random $d$-regular directed graphs in [13].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-16T01:04:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "15B33",
      "05C80"
    ],
    "keywords": [
      "regular graphs",
      "adjacency matrices",
      "high probability",
      "local central limit theorem",
      "invertibility"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}