{
  "id": "1305.0601",
  "version": "v1",
  "published": "2013-05-03T00:25:24.000Z",
  "updated": "2013-05-03T00:25:24.000Z",
  "title": "On the Cayley graph of a commutative ring with respect to its zero-divisors",
  "authors": [
    "Ghodratollah Aalipour",
    "Saieed Akbari"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we study $\\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\\mathbb{CAY}(R)$ as well. We also study $Reg(\\mathbb{CAY}(R))$ the induced subgraph on the regular elements of $R$. This graph gives a family of vertex transitive graphs. We show that if $R$ is a Noetherian ring and $Reg(\\mathbb{CAY}(R))$ has no infinite clique, then $R$ is finite. Furthermore, for every finite ring $R$, the clique number and the chromatic number of $Reg(\\mathbb{CAY}(R))$ are determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-03T00:25:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C25",
      "05C40",
      "05C69",
      "16N40"
    ],
    "keywords": [
      "cayley graph",
      "commutative ring",
      "clique number",
      "infinite clique",
      "vertex transitive graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.0601A"
    }
  }
}