{
  "id": "1406.3223",
  "version": "v2",
  "published": "2014-06-12T12:55:40.000Z",
  "updated": "2014-11-24T16:46:59.000Z",
  "title": "Cayley-type graphs for group-subgroup pairs",
  "authors": [
    "Cid Reyes-Bustos"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "In this paper we introduce a Cayley-type graph for group-subgroup pairs and present some elementary properties of such graphs, including connectedness, their degree and partition structure, and vertex-transitivity. We relate these properties to those of the underlying group-subgroup pair. From the properties of the group, subgroup and generating set some of the eigenvalues can be determined, including the largest eigenvalue of the graph. In particular, when this construction results in a bipartite regular graph we show a sufficient condition on the size of the generating sets that results on Ramanujan graphs for a fixed group-subgroup pair. Examples of Ramanujan pair-graphs that do not satisfy this condition are also provided, to show that the condition is not necessary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-12T12:55:40.000Z",
      "title": "Group-Subgroup Pair Graphs",
      "abstract": "Following a recent development on the extension of the group determinant to group-subgroup pairs, we introduce a generalization of Cayley graphs for such pairs. We present some elementary properties of these graphs, including connectedness, their degree and partition structure, and vertex-transitivity. We relate these properties to those of the underlying group-subgroup pair. From the properties of the group, subgroup and generating set some of the eigenvalues can be determined, including the largest eigenvalue of the graph. In particular, when this construction results in a bipartite regular graph we show a sufficient condition on the size of the generating sets that results on Ramanujan graphs for a fixed group-subgroup pair.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-24T16:46:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C50"
    ],
    "keywords": [
      "group-subgroup pair graphs",
      "bipartite regular graph",
      "generating set",
      "fixed group-subgroup pair",
      "cayley graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.3223R"
    }
  }
}