Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

  • E.M. Mehdi-Nezhad
  • A.M. Rahimi
Keywords: Comaximal, ideal-based, zero-divisor, graph, dominating set, domination number

Abstract

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by ΓI (R), is the graph whose vertices are the set {xR \ I| xyI for some yR \ I} and two distinct vertices x and y are adjacent if and only if xyI. Define the comaximal graph of R, denoted by CG(R) to be a graph whose vertices are the elements of R, where two distinct vertices ɑ and b are adjacent if and only if + Rb = R. A nonempty set SV of a graph G = (V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of ΓI (R) and the comaximal graph CG2(R) \ J(R) (or CGJ (R) for short) where CG2(R) is the subgraph of CG(R) induced on the nonunit elements of R and J(R) is the Jacobson radical of R.

Keywords: Comaximal, ideal-based, zero-divisor, graph, dominating set, domination number.

Published
2016-08-11
Section
Articles

Journal Identifiers


eISSN: 1727-933X
print ISSN: 1607-3606