Extended total graph associated to finite commutative rings

UDC 512.5 For a commutative ring R   with nonzero identity 1 ≠ 0 , let Z ( R ) 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 as  Z ( R ) , and 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 .  In addition, we characterize finite commutative nonlocal rings R for which  the corresponding graph T ( Γ e ( R ) ) has the clique number 4.