آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 122 |
| تعداد مقالات | 1,544 |
| تعداد مشاهده مقاله | 1,636,698 |
| تعداد دریافت فایل اصل مقاله | 1,534,197 |
On the essential dot product graph of a commutative ring | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 5، دوره 10، شماره 2، شهریور 2025، صفحه 319-334 اصل مقاله (430.94 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28166.1465 | ||
| نویسندگان | ||
| Asma Ali؛ Bakhtiyar Ahmad* | ||
| Mathematics, Science, Aligarh Muslim University, Aligarh, India | ||
| چکیده | ||
| Let $\mathcal{B}$ be a commutative ring with unity $1\neq 0$, $1\leq m <\infty$ be an integer and $\mathcal{R}=\mathcal{B}\times \mathcal{B} \times\cdots\times \mathcal{B}$ ($m$ times). The total essential dot product graph $ETD(\mathcal{R})$ and the essential zero-divisor dot product graph $EZD(\mathcal{R})$ are undirected graphs with the vertex sets $\mathcal{R}^{*} = \mathcal{R}\setminus \{(0,0,...0)\}$ and $Z(\mathcal{R})^*=Z(\mathcal{R})\setminus \{(0,0,...,0)\}$ respectively. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are adjacent if and only if $ann_\mathcal{B}(w\cdot z)$ is an essential ideal of $\mathcal{B}$ (where $w\cdot z=w_1z_1+w_2z_2+\cdots +w_mz_m\in \mathcal{B}$). In this paper, we prove some results on connectedness, diameter and girth of $ETD(\mathcal{R})$ and $EZD(\mathcal{R})$. We classify the ring $\mathcal{R}$ such that $EZD(\mathcal{R})$ and $ETD(\mathcal{R})$ are planar, outerplanar, and of genus one. | ||
| کلیدواژهها | ||
| essential graph؛ dot product graph؛ planar graph؛ genus؛ reduced ring؛ essential ideal | ||
| مراجع | ||
|
[1] S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), no. 2, 847–855. https://doi.org/10.1016/S0021-8693(03)00435-6
[2] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 296 (2006), no. 2, 462–479. https://doi.org/10.1016/j.jalgebra.2005.07.007
[3] D.F. Anderson, R. Levy, and J. Shapiro, Zero-divisor graphs, von neumann regular rings, and boolean algebras, J. Pure Appl. Algebra 180 (2003), no. 3, 221–241. https://doi.org/10.1016/S0022-4049(02)00250-5
[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 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
[6] A. Azadi, M.J. Nikmehr, and B. Soleymanzadeh, Perfect essential graphs, Le Matematiche 74 (2019), no. 2, 279–287.
[7] A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra 43 (2015), no. 1, 43–50. https://doi.org/10.1080/00927872.2014.897188
[8] J. Battle, F. Harary, and Y. Kodama, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (1962), no. 6, 565–568.
[9] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
[10] J.D. LaGrange, Complemented zero-divisor graphs and boolean rings, J. Algebra 315 (2007), no. 2, 600–611. https://doi.org/10.1016/j.jalgebra.2006.12.030
[11] T.G. Lucas, The diameter of a zero divisor graph, J. Algebra 301 (2006), no. 1, 174–193. https://doi.org/10.1016/j.jalgebra.2006.01.019
[12] M. Nazim and N. ur Rehman, On the essential annihilating-ideal graph of commutative rings., Ars Math. Contemp. 22 (2022), no. 3, #P3.05. https://doi.org/10.26493/1855-3974.2645.8fc
[13] M.J. Nikmehr, R. Nikandish, and M. Bakhtyiari, On the essential graph of a commutative ring, J. Algebra Appl. 16 (2017), no. 7, Article ID: 1750132. https://doi.org/10.1142/S0219498817501328
[14] S. Payrovi, S. Babaei, and E.S. Sevim, On the compressed essential graph of a commutative ring, Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 3, 421–429. https://doi.org/10.36045/bbms/1568685656
[15] K. Selvakumar, M. Subajini, and M.J. Nikmehr, On the genus of essential graph of commutative rings., Australas. J Comb. 74 (2019), no. 1, 74–85.
[16] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, New Jersey, 1996.
[17] A.T. White, Graphs, Groups and Surfaces, Elsevier, New York, 1985. | ||
|
آمار تعداد مشاهده مقاله: 542 تعداد دریافت فایل اصل مقاله: 1,978 |
||