On the Genus of the Total Graph of a Commutative Ring

Abstract

Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by TΓ(R), is the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Maimani et al. [13] determined all isomorphism classes of finite commutative rings whose total graph has genus at most one. In this article, after enumerating certain lower and upper bounds for genus of the total graph of a commutative ring, we characterize all isomorphism classes of finite commutative rings whose total graph has genus two.