{
  "id": "1301.1452",
  "version": "v1",
  "published": "2013-01-08T09:16:35.000Z",
  "updated": "2013-01-08T09:16:35.000Z",
  "title": "Independent sets of some graphs associated to commutative rings",
  "authors": [
    "Saeid Alikhani",
    "Saeed Mirvakili"
  ],
  "comment": "27 pages. 22 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G=(V,E)$ be a simple graph. A set $S\\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with cardinality Let $R$ be a commutative ring with nonzero identity and $I$ an ideal of $R$. The zero-divisor graph of $R$, denoted by $\\Gamma(R)$, is an undirected graph whose vertices are the nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. Also the ideal-based zero-divisor graph of $R$, denoted by $\\Gamma_I(R)$, is the graph which vertices are the set ${x\\in R\\backslash I | xy\\in I \\quad for some \\quad y\\in R\\backslash I\\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy \\in I$. In this paper we study the independent sets and the independence number of $\\Gamma(R)$ and $\\Gamma_I(R)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-08T09:16:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "13A99"
    ],
    "keywords": [
      "commutative ring",
      "distinct vertices",
      "independence number",
      "maximum independent set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.1452A"
    }
  }
}