Topological Graph-Theoretic Properties of the Generalized Zero-Divisor Graph of Commutative Rings

Let P be a commutative ring with identity, and denote by D(P) the collection of all zero-divisors of P. An ideal A of P is called an essential ideal if its intersection with every nontrivial ideal of P is nonzero. In such a case, we write A≤eP. The generalized zero-divisors graph of P, denoted by Γg(P), is defined as the simple undirected graph whose vertex set is D(P)∗=D(P)∖{0}. For two distinct vertices μ and σ, an edge joins them precisely when the sum of their annihilator ideals, namely Ann(μ)+Ann(σ), forms an essential ideal of P. This work begins by identifying all finite commutative rings with identity whose generalized zero-divisors graph possesses outerplanarity index equal to 2. Subsequently, we provide a complete classification of finite commutative rings P for which Γg(P) admits embeddings as a double-toroidal graph, a projective-plane graph, or a Klein-bottle graph. In addition, the book thickness of Γg(P) is established for the class of graphs with genus of at most one.