Extended total graph associated to finite commutative rings - Altaf, Aaqib; Pirzada, S.; Alghamdi, Ahmad M.; Almotairi, Eman S. (Ukrains’kyi Matematychnyi Zhurnal, 2024)
UDC 512.5For 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 $x+y\in Z(R)$. 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 $x+y\in Z^*(R)$, 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 $4.$