{
  "id": "2407.00714",
  "version": "v1",
  "published": "2024-06-30T14:34:33.000Z",
  "updated": "2024-06-30T14:34:33.000Z",
  "title": "On $Q$-Polynomial distance-regular graphs with a linear dependency involving a $3$-clique",
  "authors": [
    "Mojtaba Jazaeri"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Gamma$ denote a distance-regular graph with diameter $D \\geq 2$. Let $E$ denote a primitive idempotent of $\\Gamma$ with respect to which $\\Gamma$ is $Q$-polynomial. Assume that there exists a $3$-clique $x,y,z$ such that $E\\hat{x},E\\hat{y},E\\hat{z}$ are linearly dependent. In this paper, we classify all the $Q$-polynomial distance-regular graphs $\\Gamma$ with the above property. We describe these graphs from multiple points of view.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-30T14:34:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial distance-regular graphs",
      "linear dependency",
      "multiple points",
      "linearly dependent",
      "primitive idempotent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}