TY - JOUR
ID - 14651
TI - Cliques in the extended zero-divisor graph of finite commutative rings
JO - Communications in Combinatorics and Optimization
JA - CCO
LA - en
SN - 2538-2128
AU - Pirzada, Shariefuddin
AU - Altaf, Aaqib
AD - Department of Mathematics, University of Kashmir, Srinagar, India
AD - Department of Mathematics, Lovely Professional University, Punjab, India
Y1 - 2023
PY - 2023
VL -
IS -
SP -
EP -
KW - Zero-divisor graph
KW - Extended zero-divisor graph
KW - finite commutative rings
KW - clique number
DO - 10.22049/cco.2023.28740.1693
N2 - Let $R$ be a finite commutative ring with or without unity and $\Gamma_{e}(R)$ be its extended zero-divisor graph with vertex set $Z^{*}(R)=Z(R)\setminus \lbrace0\rbrace$ and two distinct vertices $x,y$ are adjacent if and only if $x.y=0$ or $x+y\in Z^{*}(R)$. In this paper, we characterize finite commutative rings whose extended zero-divisor graph have clique number $1 ~ \text{or}~ 2$. We completely characterize the rings of the form $R\cong R_1\times R_2 $, where $R_1$ and $R_2$ are local, having clique number $3,~4~\text{or}~5$. Further we determine the rings of the form $R\cong R_1\times R_2 \times R_3$, where $R_1$,$R_2$ and $R_3$ are local rings, to have clique number equal to six.
UR - https://comb-opt.azaruniv.ac.ir/article_14651.html
L1 - https://comb-opt.azaruniv.ac.ir/article_14651_e70cc7687ea0a7593283a2693d0f51f1.pdf
ER -