Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings

Abstract

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by ZA. An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A. It is denoted by ℐ≤eA. The generalized zero-divisor graph denoted by ΓgA is an undirected graph with vertex set ZA∗ (set of all nonzero zero-divisors of A) and two distinct vertices x1 and x2 are adjacent if and only if annx1+annx2≤eA. In this paper, first we characterized all the finite commutative rings A for which ΓgA is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which ΓgA is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of ΓgA.