Extended total graph associated to finite commutative rings - Altaf, Aaqib; Pirzada, S.; Alghamdi, Ahmad M.; Almotairi, Eman S. (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 512.5 For a commutative ring R with nonzero identity 1 eq 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+yZ(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+yZ^*(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.