{
  "id": "1701.03193",
  "version": "v1",
  "published": "2017-01-12T00:14:51.000Z",
  "updated": "2017-01-12T00:14:51.000Z",
  "title": "Partially metric association schemes with a multiplicity three",
  "authors": [
    "Edwin R. van Dam",
    "Jack H. Koolen",
    "Jongyook Park"
  ],
  "comment": "26 pages, 12 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete $4$-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known $2$-arc-transitive covers of the cube: the M\\\"{o}bius-Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric association schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive $2$-walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-12T00:14:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05C50"
    ],
    "keywords": [
      "multiplicity",
      "symmetric partially metric association schemes",
      "symmetric association schemes",
      "foster graph f048a",
      "connected relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}