Rings whose total graphs have genus at most one
Let R be a commutative ring with 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 ε R, the vertices x and y are adjacent if and only if x + y ε 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.
Genus, Planar graph, Toroidal graph, Total graph
Maimani, Hamid Reza, Cameron Wickham, and Siamak Yassemi. "Rings whose total graphs have genus at most one." The Rocky Mountain Journal of Mathematics 42, no. 5 (2012): 1551-1560.
Rocky Mountain Journal of Mathematics