{
  "id": "2109.13849",
  "version": "v2",
  "published": "2021-09-28T16:26:17.000Z",
  "updated": "2022-03-28T19:02:11.000Z",
  "title": "On bipartite distance-regular Cayley graphs with small diameter",
  "authors": [
    "Edwin R. van Dam",
    "Mojtaba Jazaeri"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley graph which contains the identity element -- and $\\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group. We also extend a result by Miklavi\\v{c} and Poto\\v{c}nik that relates difference sets to bipartite distance-regular Cayley graphs with diameter three to the case of diameter four. This new case involves certain partial geometric difference sets and -- in the antipodal case -- relative difference sets.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-28T19:02:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "small diameter",
      "bipartite cayley graph",
      "study bipartite distance-regular cayley graphs",
      "sufficient conditions",
      "partial geometric difference sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}