{
  "id": "1401.6816",
  "version": "v1",
  "published": "2014-01-27T11:58:06.000Z",
  "updated": "2014-01-27T11:58:06.000Z",
  "title": "Strongly regular graphs with the 7-vertex condition",
  "authors": [
    "Sven Reichard"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $t$-vertex condition, for an integer $t\\ge 2$, was introduced by Hestenes and Higman in 1971, providing a combinatorial invariant defined on edges and non-edges of a graph. Finite rank 3 graphs satisfy the condition for all values of $t$. Moreover, a long-standing conjecture of M. Klin asserts the existence of an integer $t_0$ such that a graph satisfies the $t_0$-vertex condition if and only if it is a rank 3 graph. We construct the first infinite family of non-rank 3 strongly regular graphs satisfying the $7$-vertex condition. This implies that the Klin parameter $t_0$ is at least 8. The examples are the point graphs of a certain family of generalised quadrangles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-27T11:58:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30"
    ],
    "keywords": [
      "strongly regular graphs",
      "vertex condition",
      "graphs satisfy",
      "point graphs",
      "klin asserts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}