{
  "id": "2109.12805",
  "version": "v3",
  "published": "2021-09-27T05:32:19.000Z",
  "updated": "2021-11-28T16:51:27.000Z",
  "title": "Classification of divisible design graphs with at most 39 vertices",
  "authors": [
    "Dmitry Panasenko",
    "Leonid Shalaginov"
  ],
  "doi": "10.1002/jcd.21818",
  "categories": [
    "math.CO"
  ],
  "abstract": "A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\\lambda_1$ common neighbors, and two vertices from different classes have exactly $\\lambda_2$ common neighbors. A DDG with $m = 1$, $n = 1$, or $\\lambda_1 = \\lambda_2$ is called improper, otherwise it is called proper. We present new constructions of DDGs and, using a computer enumeration algorithm, we find all proper connected DDGs with at most 39 vertices, except for three tuples of parameters: $(32,15,6,7,4,8)$, $(32,17,8,9,4,8)$, $(36,24,15,16,4,9)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2021-11-28T16:51:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divisible design graph",
      "classification",
      "common neighbors",
      "computer enumeration algorithm",
      "vertex set"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}