Rings whose total graphs have genus at most one

Let $R$ be a commutative ring with $\mathbb{Z}(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\text{T}(\mathrm{\Gamma }(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in R$,thevertices$x$and$y$areadjacentifandonlyif$x + y\in\Z®$. We investigate properties of the total graph of $R$ and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer $g$, there are only finitely many finite rings whose total graph has genus $g$.