{
  "id": "1505.06010",
  "version": "v1",
  "published": "2015-05-22T09:55:32.000Z",
  "updated": "2015-05-22T09:55:32.000Z",
  "title": "Optimal extensions and quotients of 2--Cayley Digraphs",
  "authors": [
    "F. Aguiló",
    "A. Miralles",
    "M. Zaragozá"
  ],
  "comment": "15 pages, 4 tables, 3 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "Given a finite Abelian group $G$ and a generator subset $A\\subset G$ of cardinality two, we consider the Cayley digraph $\\Gamma=$Cay$(G,A)$. This digraph is called $2$--Cayley digraph. An extension of $\\Gamma$ is a $2$--Cayley digraph, $\\Gamma'=$Cay$(G',A)$ with $G<G'$, such that there is some subgroup $H<G'$ satisfying the digraph isomorphism Cay$(G'/H,A)\\cong$Cay$(G,A)$. We also call the digraph $\\Gamma$ a quotient of $\\Gamma'$. Notice that the generator set does not change. A $2$--Cayley digraph is called optimal when its diameter is optimal with respect to its order. In this work we define two procedures, E and Q, which generate a particular type of extensions and quotients of $2$--Cayley digraphs, respectively. These procedures are used to obtain optimal quotients and extensions. Quotients obtained by procedure Q of optimal $2$--Cayley digraphs are proved to be also optimal. The number of tight extensions, generated by procedure E from a given tight digraph, is characterized. Tight digraphs for which procedure E gives infinite tight extensions are also characterized. Finally, these two procedures allow the obtention of new optimal families of $2$--Cayley digraphs and also the improvement of the diameter of many proposals in the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-22T09:55:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "cayley digraph",
      "optimal extensions",
      "tight digraph",
      "infinite tight extensions",
      "finite abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}