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Clean Graphs and Idempotent Graphs over Finite Rings: An Approach Based on $\mathbb{Z}_n$ | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 18 اردیبهشت 1405 اصل مقاله (475.68 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31307.2811 | ||
| نویسندگان | ||
| Felicia Servina Djuang؛ Indah Emilia Wijayanti* ؛ Yeni Susanti | ||
| Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia | ||
| چکیده | ||
| Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the subgraph of $Cl(R)$ induced by the set $\{(e, u): e \text{ is a nonzero idempotent element of } R\}$. In this study, we examine the structure of clean graphs over $\mathbb{Z}_{n}$ derived from their $Cl_2$ graphs and investigate their relationship with the structure of their idempotent graphs. Furthermore, we obtain an equivalence between the isomorphism of two clean graphs and the isomorphism of their corresponding idempotent graphs over an Artinian ring. | ||
| کلیدواژه‌ها | ||
| Clean graph؛ idempotent graph؛ isomorphism graph؛ local ring؛ Artinian ring | ||
| مراجع | ||
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[1] S. Akbari, M. Habibi, A. Majidinya, and R. Manaviyat, On the idempotent graph of a ring, J. Algebra Appl. 12 (2013), no. 6, 1350003. https://doi.org/10.1142/S0219498813500035
[2] D.F. Anderson, M.C. Axtell, and J.A. Stickles Jr, Zero-divisor graphs in commutative rings, 2010, pp. 23–45. https://doi.org/10.1007/978-1-4419-6990-3_2
[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] F.S. Djuang, I.E. Wijayanti, and Y. Susanti, Clean graphs and idempotent graphs over finite rings: An approach based on Z n, 2025, pp. 1–21. arXiv preprint arXiv:2505.14249
[5] F.S. Djuang, I.E. Wijayanti, and Y. Susanti, Isomorphism of clean graphs over Zn and structural insight into $M_2(\mathhb{Z}_p)$, 2025, pp. 1–13. arXiv preprint arXiv:2509.12004
[6] D.S. Dummit and R.M. Foote, Abstract Algebra, vol. 3, Wiley Hoboken, 2004.
[7] W. Fish, J.D. Key, and E. Mwambene, Codes from incidence matrices and line graphs of hamming graphs, Discrete Math. 310 (2010), no. 13-14, 1884–1897. https://doi.org/10.1016/j.disc.2010.02.010
[8] R.P. Grimaldi, Discrete and Combinatorial Mathematics, 5/e, Pearson Education India, 2006.
[9] M. Habibi, E. Yetkın C¸elıkel, and C. Abdıoğlu, Clean graph of a ring, J. Algebra Appl. 20 (2021), no. 9, 2150156. https://doi.org/10.1142/S0219498821501565 [10] N.A. Immormino, Clean Rings and Clean Group Rings, Bowling Green State University, 2013.
[11] R.S. Jain, B.S. Reddy, and W.M. Shaikh, Construction of linear codes from the unit graph G(Zn), Asian-Eur. J. Math. 16 (2023), no. 11, 2350213. https://doi.org/10.1142/S1793557123502133
[12] D.S. Malik, J.N. Mordeson, and M.K. Sen, Introduction to Abstract Algebra,Creighton University, 2007.
[13] W.K. Nicholson and Y. Zhou, Clean rings: A survey, Advances in Ring Theory, World Scientific, 2005, pp. 181–198. https://doi.org/10.1142/9789812701671_0017 [14] A. Patil, P. S. Momale, and C. M. Jadhav, On the idempotent graph of matrix ring, 2024, arXiv:2402.11256.
[15] T. Pongthana, K. Jantarakhajorn, and B. Khuhirun, Isomorphism classes of clean graphs of rings of integers modulo $n$, Tech. report, Thammasat University, 2022.
[16] R. Singh, M. Habibi, and S.C. Patekar, On the clean graph of commutative Artinian rings, Int. J. Math. Math. Sci. 2025 (2025), no. 1, 8677973. https://doi.org/10.1155/ijmm/8677973
[17] Y. Susanti, A. Sutjijana, U. Isnaini, A.S. Bawana, and C.P.L.J. Hatmakelana, Unit regular graphs over finite rings, J. Algebra Comb. Discrete Struct. Appl. 12 (2025), no. 2, 79–96. https://doi.org/10.13069/jacodesmath.v12i2.312
[18] R. J. Wilson, Introduction to Graph Theory, 5th ed., Pearson/Prentice Hall, Harlow, England, 2010. | ||
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آمار تعداد مشاهده مقاله: 221 تعداد دریافت فایل اصل مقاله: 95 |
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