Rings whose total graphs have genus at most one

Abstract

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 $\T(Γ(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x,y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x+y\in \mathbb{Z}(R)$. 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 $g$.

7 pages. To appear in Rocky Mountain Journal of Mathematics