Abstract
Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by Z(A). 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 <=(e) A. The generalized zero-divisor graph denoted by Gamma(g)(A) is an undirected graph with vertex set Z(A)* (set of all nonzero zero-divisors of A) and two distinct vertices x(1) and x(2) are adjacent if and only if ann(x(1)) + ann(x(2)) <=(e) A. In this paper, first we characterized all the finite commutative rings A for which Gamma(g)(A) is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which Gamma(g)(A) is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of Gamma(g)(A).