{
  "id": "1510.01697",
  "version": "v1",
  "published": "2015-10-06T18:48:02.000Z",
  "updated": "2015-10-06T18:48:02.000Z",
  "title": "Large $\\{0, 1, \\ldots, t\\}$-Cliques in Dual Polar Graphs",
  "authors": [
    "Ferdinand Ihringer",
    "Klaus Metsch"
  ],
  "comment": "30 pages including appendix",
  "categories": [
    "math.CO"
  ],
  "abstract": "We investigate $\\{0, 1, \\ldots, t \\}$-cliques of generators on dual polar graphs of finite classical polar spaces of rank $d$. These cliques are also known as Erd\\H{o}s-Ko-Rado sets in polar spaces of generators with pairwise intersections in at most codimension $t$. Our main result is that we classify all such cliques of maximum size for $t \\leq \\sqrt{8d/5}-2$ if $q \\geq 3$, and $t \\leq \\sqrt{8d/9}-2$ if $q = 2$. We have the following byproducts. (a) For $q \\geq 3$ we provide estimates of Hoffman's bound on these $\\{0, 1, \\ldots, t \\}$-cliques for all $t$. (b) For $q \\geq 3$ we determine the largest, second largest, and smallest eigenvalue of the graphs which have the generators of a polar space as vertices and where two generators are adjacent if and only if they meet in codimension at least $t+1$. Furthermore, we provide nice explicit formulas for all eigenvalues of these graphs. (c) We provide upper bounds on the size of the second largest maximal $\\{0, 1, \\ldots, t \\}$-cliques for some $t$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-06T18:48:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E20",
      "05B25",
      "52C10"
    ],
    "keywords": [
      "dual polar graphs",
      "generators",
      "finite classical polar spaces",
      "nice explicit formulas",
      "second largest maximal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}