{
  "id": "1806.03899",
  "version": "v1",
  "published": "2018-06-11T10:51:30.000Z",
  "updated": "2018-06-11T10:51:30.000Z",
  "title": "On solid density of Cayley digraphs on finite Abelian groups",
  "authors": [
    "F. Aguiló",
    "M. Zaragozá"
  ],
  "comment": "13 pages and 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Gamma=$Cay$(G,T)$ be a Cayley digraph over a finite Abelian group $G$ with respect the generating set $T\\not\\ni0$. $\\Gamma$ has order ord$(\\Gamma)=|G|=n$ and degree deg$(\\Gamma)=|T|=d$. Let $k(\\Gamma)$ be the diameter of $\\Gamma$ and denote $\\kappa(d,n)=\\min\\{k(\\Gamma):~\\textrm{ord}(\\Gamma)=n,\\textrm{deg}(\\Gamma)=d\\}$. We give a closed expression, $\\ell(d,n)$, of a tight lower bound of $\\kappa(d,n)$ by using the so called {\\em solid density} introduced by Fiduccia, Forcade and Zito. A digraph $\\Gamma$ of degree $d$ is called {\\em tight} when $k(\\Gamma)=\\kappa(d,|\\Gamma|)=\\ell(d,|\\Gamma|)$ holds. Recently, the {\\em Dilating Method} has been developed to derive a sequence of digraphs of constant solid density. In this work, we use this method to derive a sequence of tight digraphs $\\{\\Gamma_i\\}_{i=1}^{\\textrm{c}(\\Gamma)}$ from a given tight digraph $\\Gamma$. Moreover, we find a closed expression of the cardinality c$(\\Gamma)$ of this sequence. It is perhaps surprising that c$(\\Gamma)$ depends only on $n$ and $d$ and not on the structure of $\\Gamma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-11T10:51:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "finite abelian group",
      "cayley digraph",
      "tight digraph",
      "tight lower bound",
      "closed expression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}