{
  "id": "2106.08677",
  "version": "v1",
  "published": "2021-06-16T10:31:01.000Z",
  "updated": "2021-06-16T10:31:01.000Z",
  "title": "Divisible design graphs with parameters $(4n,n+2,n-2,2,4,n)$ and $(4n,3n-2,3n-6,2n-2,4,n)$",
  "authors": [
    "Leonid Shalaginov"
  ],
  "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. $4\\times n$-lattice graph is the line graph of $K_{4,n}$. This graph is a DDG with parameters $(4n,n+2,n-2,2,4,n)$. In the paper we consider DDGs with these parameters. We prove that if $n$ is odd then such graph can only be a $4\\times n$-lattice graph. If $n$ is even we characterise all DDGs with such parameters. Moreover, we characterise all DDGs with parameters $(4n,3n-2,3n-6,2n-2,4,n)$ which are related to $4\\times n$-lattice graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-16T10:31:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divisible design graph",
      "parameters",
      "lattice graph",
      "common neighbors",
      "characterise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}