Extended total graph associated to finite commutative rings

Abstract

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.