TY - JOUR
AU - Aaqib Altaf
AU - S. Pirzada
AU - Ahmad M. Alghamdi
AU - Eman S. Almotairi
PY - 2024/07/03
Y2 - 2024/07/21
TI - Extended total graph associated to finite commutative rings
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 76
IS - 6
SE - Research articles
DO - 10.3842/umzh.v76i5.7494
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/7494
AB - 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 $T_{\Gamma}(R)$ 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 $x+y\in Z(R)$. In this paper, we define an extension of the total graph denoted by $T(\Gamma^{e}(R))$ 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 $T(\Gamma^{e}(R))$ has clique numbers $1,2,$ and $3$. In addition, we characterize finite commutative nonlocal rings $R$ for which the corresponding graph $T(\Gamma^{e}(R))$ has the clique number $4.$
ER -