{
  "id": "2105.11787",
  "version": "v1",
  "published": "2021-05-25T09:27:53.000Z",
  "updated": "2021-05-25T09:27:53.000Z",
  "title": "Quasi-strongly regular graphs of grade three with diameter two",
  "authors": [
    "Songpon Sriwongsa",
    "Pawaton Kaemawichanurat"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A quasi-strongly regular graph of grade $p$ with parameters $(n, k, a; c_1, \\ldots, c_p)$ is a $k$-regular graph of order $n$ such that any two adjacent vertices share $a$ common neighbours and any two non-adjacent vertices share $c_{i}$ common neighbours for some $1 \\leq i \\leq p$. This is a generalization of a strongly regular graph. In this paper, we focus on strictly quasi-strongly regular graphs of grade $3$ with $c_i = k - i$ for $i = 1, 2, 3$. The main result is to show the sharp bounds of order $n$ for a given $k \\geq 4$. Furthermore, by this result, we characterize all of these graphs whose $n$ satisfies upper or lower bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-25T09:27:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05C75"
    ],
    "keywords": [
      "common neighbours",
      "non-adjacent vertices share",
      "lower bounds",
      "satisfies upper",
      "sharp bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}