Extended Total Graph Associated with Finite Commutative Rings

For a commutative ring R with nonzero identity 1 ≠ 0, by Z(R) we denote the set of zero divisors. The total graph of R denoted by T_Γ(R) is a simple graph in which all elements of R are vertices and any two distinct vertices x and y are adjacent if and only if x+y ∈ Z(R). In this paper, we define an extension of the total graph denoted by T(Γ^e(R)) with vertex set Z(R) in which two distinct vertices x and y are adjacent if and only if x + y ∈ Z^*(R), where Z^* (R) is the set of nonzero zero divisors of R. Our main aim is to characterize the finite commutative rings whose T(Γ^e(R)) has clique numbers 1, 2, and 3. Moreover, we characterize finite commutative nonlocal rings R for which the corresponding graph T(Γ^e(R)) has the clique number 4.