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Cliques in the extended zero-divisor graph of finite commutative rings | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 12، دوره 10، شماره 1، خرداد 2025، صفحه 195-206 اصل مقاله (436.37 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28740.1693 | ||
| نویسندگان | ||
| Shariefuddin Pirzada* 1؛ Aaqib Altaf2 | ||
| 1Department of Mathematics, University of Kashmir, Srinagar, India | ||
| 2Department of Mathematics, Lovely Professional University, Punjab, India | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| zero-divisor graph؛ extended zero-divisor graph؛ finite commutative rings؛ clique number | ||
| مراجع | ||
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