Extended Total Graph Associated with Finite Commutative Rings

For a commutative ring R with nonzero identity 1 not equal 0, by Z(R) we denote the set of zero divisors. The total graph of R denoted by T Gamma(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 is an element of Z(R). In this paper, we define an extension of the total graph denoted by T(Gamma e(R)) with vertex set Z(R) in which two distinct vertices x and y are adjacent if and only if x + y is an element of 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(Gamma e(R)) has clique numbers 1, 2, and 3. Moreover, we characterize finite commutative nonlocal rings R for which the corresponding graph T(Gamma e(R)) has the clique number 4.