Total Graphs Associated to a Commutative Ring,

Abstract

Let R be a commutative ring with nonzero unity. Let Z(R) be the set of all zerodivisors of R. The total graph of R, denoted by T (Γ(R)), is the simple graph with vertex set R and two distinct vertices x and y are adjacent if their sum x + y ∈ Z(R). Several authors presented various generalizations for T (Γ(R)). This article surveys research conducted on T (Γ(R)) and its generalizations. A historical review of literature is given. Further properties of T (Γ(R)) are also studied. Many open problems are presented for further research.