آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 122 |
| تعداد مقالات | 1,537 |
| تعداد مشاهده مقاله | 1,622,449 |
| تعداد دریافت فایل اصل مقاله | 1,522,373 |
On cozero divisor graphs of ring $Z_n$ | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 16، دوره 11، شماره 1، خرداد 2026، صفحه 227-245 اصل مقاله (472.77 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.26112.1974 | ||
| نویسندگان | ||
| Zahid Raza1؛ Bilal Ahmad Rather2؛ Modjtaba Ghorbani* 3 | ||
| 1Department of Mathematics, College of Sciences, University of Sharjah, UAE | ||
| 2Mathematical Sciences Department, College of Science, United Arab Emirates University, Al Ain 15551, Abu Dhabi, UAE | ||
| 3Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-163, I.R. Iran | ||
| چکیده | ||
| The cozero divisor graph $\Gamma^{\prime}(R)$ of a commutative ring $R$ is a simple graph with vertex set as non-zero zero divisor elements of $R$ such that two distinct vertices $x$ and $y$ are adjacent iff $x\notin Ry$ and $y\notin Rx$, where $xR$ is the ideal generated by $x$. In this article we find the spectra of $\Gamma^{\prime}(\mathbb{Z}_{n}) $ for $n\in \{q_{1}q_{2}, q_{1}q_{2}q_{3},q_{1}^{n_{1}}q_{2}\},$ where $q_{i}$'s are primes. As a consequence we obtain the bounds for the largest (smallest) eigenvalues, bounds for spread, rank and inertia of $ \Gamma^{\prime}(\mathbb{Z}_{q_{1}^{n_{1}}q_{2}})$ along with the determinant, inverse and square of trace of its quotient matrix. We present the extremal bounds for the energy of $\Gamma^{\prime}(\mathbb{Z}_{n})$ for $n=q_{1}^{n_{1}}q_{2}$ and characterize the extremal graphs attaining them. We close article with conclusion for furtherance. | ||
| کلیدواژهها | ||
| spectra؛ energy؛ cozero divisor graphs؛ commutative rings | ||
| مراجع | ||
|
[1] M. Afkhami and K. Khashyarmanesh, The cozero divisor graph of a commutative ring, Southeast Asian Bull. Math. 35 (2011), no. 5, 753–762.
[2] M. Afkhami and K. Khashyarmanesh, On the cozero-divisor graphs of commutative rings and their complements, Bull. Malays. Math. Sci. Soc. 35 (2012), no. 4, 935–944.
[3] M. Afkhami and K. Khashyarmanesh, Planar, outerplanar, and ring graph of the cozero-divisor graph of a finite commutative ring, J. Algebra Appl. 11 (2012), no. 6, Article ID: 1250103. https://doi.org/10.1142/S0219498812501034 [4] M. Afkhami and K. Khashyarmanesh, On the cozero-divisor graphs and comaximal graphs of commutative rings, J. Algebra Appl. 12 (2013), no. 3, Article ID: 1250173. https://doi.org/10.1142/S0219498812501733
[5] S. Akbari, F. Alizadeh, and S. Khojasteh, Some results on cozero-divisor graph of a commutative ring, J. Algebra Appl. 13 (2014), no. 3, Article ID: 1350113. https://doi.org/10.1142/S0219498813501132
[6] S. Akbari and S. Khojasteh, Commutative rings whose cozero-divisor graphs are unicyclic or of bounded degree, Comm. Algebra 42 (2014), no. 4, 1594–1605.
[7] M. Bakhtyiari, R. Nikandish, and M.J. Nikmehr, Coloring of cozero-divisor graphs of commutative von Neumann regular rings, Proc. Math. Sci. 130 (2020), no. 1, Article ID: 49. https://doi.org/10.1007/s12044-020-00569-5
[8] D.M. Cvetković, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2009.
[9] K.C. Das and P. Kumar, Some new bounds on the spectral radius of graphs, Discrete Math. 281 (2004), no. 1-3, 149–161. https://doi.org/10.1016/j.disc.2003.08.005
10] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forsch. Graz 103 (1978), 1–22.
[11] H. Kober, On the arithmetic and geometric means and on Hölder’s inequality, Proceedings of the American Mathematical Society 9 (1958), no. 3, 452–459. https://doi.org/10.2307/2033003
[12] X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, 2012.
[13] P. Mathil, B. Baloda, and J. Kumar, On the cozero-divisor graphs associated to rings, AKCE Int. J. Graphs Comb. 19 (2022), no. 3, 238–248. https://doi.org/10.1080/09728600.2022.2111241
[14] R. Nikandish, M.J. Nikmehr, and M. Bakhtyiari, Metric and strong metric dimension in cozero-divisor graphs, Mediterr. J. Math. 18 (2021), no. 3, Article ID: 112. https://doi.org/10.1007/s00009-021-01772-y
[15] S. Shen, W. Liu, and W. Jin, Laplacian eigenvalues of the unit graph of the ring Zn, Appl. Math. Comput. 459 (2023), Article ID: 128268. https://doi.org/10.1016/j.amc.2023.128268
[16] M. Young, Adjacency matrices of zero-divisor graphs of integers modulo n, Involve 8 (2015), no. 5, 753–761. https://doi.org/10.2140/involve.2015.8.753 | ||
|
آمار تعداد مشاهده مقاله: 730 تعداد دریافت فایل اصل مقاله: 995 |
||