{
  "id": "1609.06022",
  "version": "v1",
  "published": "2016-09-20T05:28:52.000Z",
  "updated": "2016-09-20T05:28:52.000Z",
  "title": "Expander property of the Cayley Graphs of $\\mathbb{Z}_m \\ltimes \\mathbb{Z}_n$",
  "authors": [
    "Kashyap Rajeevsarathy",
    "S. Lakshmivarahan",
    "Pawan Kumar Arora"
  ],
  "comment": "11 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{T}_{m,n,k}$ denote the Cayley Graph of the split metacyclic group $\\mathbb{Z}_m \\ltimes_k \\mathbb{Z}_n$ with respect to the symmetric generating set $S = \\{(\\pm,0),(0,\\pm 1)\\}$. In this paper, we derive conditions for $\\mathcal{T}_{m,n,k}$ to be a Ramanujan graph. In particular, we establish that $\\mathcal{T}_{m,n,k}$ is not Ramanujan when $\\min(m,n) > 8$, and hence, there are only finitely many Ramanujan graphs in the family $\\{\\mathcal{T}_{m,n,k}\\}$. We explicitly list all possible $(m,n,k)$ for which $\\mathcal{T}_{m,n,k}$ is Ramanujan. We conclude that the graphs in the family $\\{\\mathcal{T}_{m,n,k}\\}$ are not good spectral expanders for large $m\\text{ and } n$, which make them unsuitable for adaptation into large communication networks. Finally, we draw similar conclusions for the Cayley graphs of finite abelian groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-20T05:28:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R10",
      "05C50"
    ],
    "keywords": [
      "cayley graph",
      "expander property",
      "ramanujan graph",
      "finite abelian groups",
      "draw similar conclusions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}