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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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[1] S. Akbari and A. Mohammadian, On zero-divisor graphs of finite rings, J. Algebra 314 (2007), no. 1, 168–184. https://doi.org/10.1016/j.jalgebra.2007.02.051
[2] A.S. Alali, a. Shahbaz, H. Noor, A.M. Mahnashi, Y. Shang, and A. Assiry, Algebraic structure graphs over the commutative ring $\mathbb{Z}_m$: Exploring topological indices and entropies using $m$-polynomials, Mathematics 11 (2023), no. 18, Article ID: 3833. https://doi.org/10.3390/math11183833
[3] D.F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706–2719. https://doi.org/10.1016/j.jalgebra.2008.06.028
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447. https://doi.org/10.1006/jabr.1998.7840
[5] D.F. Anderson and D. Weber, The zero-divisor graph of a commutative ring without identity, Int. Electron. J. Algebra 23 (2018), no. 23, 176–202. https://doi.org/10.24330/ieja.373663
[6] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
[7] D. Bennis, J. Mikram, and F. Taraza, On the extended zero divisor graph of commutative rings, Turkish J. Math. 40 (2016), no. 2, 376–388. https://doi.org/10.3906/mat-1504-61
[8] A. Cherrabi, H. Essannouni, E. Jabbouri, and A. Ouadfel, On a new extension of the zero-divisor graph, Algebra Colloq. 27 (2020), no. 3, 469–476. https://doi.org/10.1142/S1005386720000383
[9] David Steven Dummit and Richard M Foote, Abstract Algebra, vol. 3, Wiley Hoboken, 2004.
[10] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, III-London, 2006.
[11] Q. Liu, T. Wu, and J. Guo, Finite rings whose graphs have clique number less than five, 28 (2021), no. 3, 533–540. https://doi.org/10.1142/S1005386721000419
[12] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient BlackSwan, Hyderabad, 2012.
[13] S.P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math. 307 (2007), no. 9-10, 1155–1166. https://doi.org/10.1016/j.disc.2006.07.025 | ||
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