{
  "id": "1509.04574",
  "version": "v1",
  "published": "2015-09-15T14:17:33.000Z",
  "updated": "2015-09-15T14:17:33.000Z",
  "title": "Intersection graph of cyclic subgroups of groups",
  "authors": [
    "R. Rajkumar",
    "P. Devi"
  ],
  "comment": "10 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\\mathscr I_c(G)$ are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graph of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. We show that for a given finite group $G$, $girth(\\mathscr I_c (G)) \\in \\{3, \\infty\\}$. Moreover, we classify all finite non-cyclic abelian groups whose intersection graph of cyclic subgroups is planar. Also for any group $G$, we determine the independence number, clique cover number of $\\mathscr I_c (G)$ and show that $\\mathscr I_c (G)$ is weakly $\\alpha$-perfect. Among the other results, we determine the values of $n$ for which $\\mathscr I_c (\\mathbb{Z}_n)$ is regular and estimate its domination number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-15T14:17:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C10",
      "05C17"
    ],
    "keywords": [
      "intersection graph",
      "finite non-cyclic abelian groups",
      "finite group",
      "proper cyclic subgroups",
      "clique cover number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150904574R"
    }
  }
}