The zero-divisor associate graph over a finite commutative ring

Biswas, Bijon; Sen Gupta, Raibatak; Sen, Mridul Kanti; Kar, Sukhendu

doi:10.22049/cco.2023.28488.1577

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	The zero-divisor associate graph over a finite commutative ring
Communications in Combinatorics and Optimization
مقاله 15، دوره 10، شماره 1، خرداد 2025، صفحه 232-243 اصل مقاله (430.38 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2023.28488.1577
نویسندگان
Bijon Biswas¹؛ Raibatak Sen Gupta^* ²؛ Mridul Kanti Sen³؛ Sukhendu Kar⁴
¹Department of Science and Humanities, Ranaghat Government Polytechnic, Nadia - 741201, WB, India
²Department of Mathematics, Bejoy Narayan Mahavidyalaya, West Bengal-712147, India
³Department of Pure Mathematics, University of Calcutta, Kolkata - 700019, India
⁴Department of Mathematics, Jadavpur University, Kolkata - 700032, India
چکیده
In this paper, we introduce the zero-divisor associate graph $\Gamma_D(R)$ over a finite commutative ring $R$. It is a simple undirected graph whose vertex set consists of all non-zero elements of $R$, and two vertices $a, b$ are adjacent if and only if there exist non-zero zero-divisors $z_1, z_2$ in $R$ such that $az_1=bz_2$. We determine the necessary and sufficient conditions for connectedness and completeness of $\Gamma_D(R)$ for a unitary commutative ring $R$. The chromatic number of $\Gamma_D(R)$ is also studied. Next, we characterize the rings $R$ for which $\Gamma_D(R)$ becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as $\Gamma_D(R)$, characterizing the associated ring $R$ in each case.
کلیدواژه‌ها
zero-divisor؛ commutative ring؛ chromatic number؛ complete graph؛ line graph

مراجع
[1] D.D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500–514. https://doi.org/10.1006/jabr.1993.1171 [2] 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 [3] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021-8693(88)90202-5 [4] G. Bini and F. Flamini, Finite Commutative Rings and Their Applications, vol. 680, Springer Science & Business Media, New York, 2002. [5] B. Biswas, S. Kar, M.K. Sen, and T.K. Dutta, A generalization of co-maximal graph of commutative rings, Discrete Math. Algorithms Appl. 11 (2019), no. 1, Atricle ID: 1950013. https://doi.org/10.1142/S1793830919500137 [6] B. Biswas, R. Sen Gupta, M.K. Sen, and S. Kar, On the connectedness of square element graphs over arbitrary rings, Southeast Asian Bulletin of Mathematics 43 (2019), no. 2, 153–164. [7] B. Biswas, R. Sen Gupta, M.K. Sen, and S. Kar, Some properties of square element graphs over semigroups, AKCE Int. J. Graphs Comb. 17 (2020), no. 1, 118–130. https://doi.org/10.1016/j.akcej.2019.02.001 [8] H.J. Chiang-Hsieh, N.O. Smith, and H.J. Wang, Commutative rings with toroidal zero-divisor graphs, Houston J. Math. 36 (2007), no. 1, 1–31. [9] J.A. Gallian, Contemporary Abstract Algebra, Houghton Mifflin, Boston, 2002. [10] M.J. González, On distinguishing local finite rings from finite rings only by counting elements and zero divisors, Eur. J. Pure Appl. Math. 7 (2014), no. 1, 109–113. [11] S.B. Nam, Finite local rings of order ≤ 16 with nonzero Jacobson radical, Korean J. Math. 21 (2013), no. 1, 23–28. https://doi.org/10.11568/kjm.2013.21.1.23 [12] D.K. Ray-Chaudhuri, Characterization of line graphs, J. Combin. Theory 3 (1967), no. 3, 201–214. https://doi.org/10.1016/S0021-9800(67)80068-1 [13] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commutative Rings 1 (2002), no. 4, 203–211. [14] 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 [15] Ľ. Šoltés, Forbidden induced subgraphs for line graphs, Discrete Math. 132 (1994), no. 1-3, 391–394. https://doi.org/10.1016/0012-365X(92)00577-E [16] S. Visweswaran and A. Parmar, Some results on the complement of a new graph associated to a commutative ring, Commun. Comb. Optim. 2 (2017), no. 2, 119–138. https://doi.org/10.22049/cco.2017.25908.1053 [17] D.B. West, Introduction to Graph Theory, Prentice Hall of India, New Delhi, 2003.
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The zero-divisor associate graph over a finite commutative ring