The total graph of a commutative ring

Let R be a commutative ring with Nil(R) its ideal of nilpotent elements, Z(R) its set of zero-divisors, and Reg(R) its set of regular elements. In this paper, we introduce and investigate 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∈R, the vertices x and y are adjacent if and only if x+y∈Z(R). We also study the three (induced) subgraphs Nil(Γ(R)), Z(Γ(R)), and Reg(Γ(R)) of T(Γ(R)), with vertices Nil(R), Z(R), and Reg(R), respectively.