Finite commutative rings whose line graphs of comaximal graphs have genus at most two

Yıl 2024, Cilt: 53 Sayı: 4, 1075 - 1084, 27.08.2024

Huadong Su , Chunhong Huang

https://doi.org/10.15672/hujms.1256413

Öz

Let $R$ be a ring with identity. The comaximal graph of $R$, denoted by $\Gamma(R)$, is a simple graph with vertex set $R$ and two different vertices $a$ and $b$ are adjacent if and only if $aR+bR=R$. Let $\Gamma_{2}(R)$ be a subgraph of $\Gamma(R)$ induced by $R\backslash\{U(R)\cup J(R)\}$. In this paper, we investigate the genus of the line graph $L(\Gamma(R))$ of $\Gamma(R)$ and the line graph $L(\Gamma_{2}(R))$ of $\Gamma_2(R)$. All finite commutative rings whose genus of $L(\Gamma(R))$ and $L(\Gamma_{2}(R))$ are 0, 1, 2 are completely characterized, respectively.

Anahtar Kelimeler

finite commutative ring, comaximal graph, line graph, genus, induced subgraph

Teşekkür

This research was supported by the Natural Science Foundation of China (Grant No. 12261001) and the Guangxi Natural Science Foundation (Grant No. 2021GXNSFAA220043) and High-level talents for scientific research of Beibu Gulf University (2020KYQD07).

Kaynakça

• [1] D.F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (7), 2706–2719, 2008.
• [2] D.F. Anderson, T. Asir, A. Badawi and T. Tamizh Chelvam, Graphs from rings, Springer, 2021.
• [3] N. Ashrafi, H.R. Maimani, M.R. Pournaki and S. Yassemi, Unit graphs associated with rings, Comm. Algebra, 38 (8), 2851–2871, 2010.
• [4] M. Azadi and Z. Jafari, Some properties of comaximal ideal graph of a commutative ring, Trans. Comb. 6 (1), 29–37, 2017.
• [5] M. Azadi, Z. Jafari and C. Eslahchi, On the comaximal ideal graph of a commutative ring, Turkish J. Math. 40 (4), 905–913, 2016.
• [6] I. Beck, Coloring of commutative rings, J. Algebra, 116 (1), 208–226, 1988.
• [7] D. Bénard, Orientable imbedding of line graphs, J. Combin. Theory Ser. B, 24 (1), 34–43, 1978.
• [8] H.J. Chiang-Hsieh, P.F. Lee and H.J. Wang, The embedding of line graphs associated to the zero-divisor graphs of commutative rings, Israel J. Math. 180 (1), 193–222, 2010.
• [9] A. Eri, Z. Pucanovi, M. Andeli, et al. Some properties of the line graphs associated to the total graph of a commutative ring, Pure and Applied Mathematics Journal, 2 (2), 51–55, 2014.
• [10] H.R. Maimani, M.R. Pournaki and S. Yassemi, Zero-divisor graph with respect to an ideal, Comm. Algebra, 34 (3), 923–929, 2006.
• [11] H.R. Maimani, M. Salimi, A. Sattari, et al. Comaximal graph of commutative rings, J. Algebra, 319 (4), 1801–1808, 2008.
• [12] S.M. Moconja and Z.Z. Petrovic, On the structure of comaximal graphs of commutative rings with identity, Bull. Aust. Math. Soc. 83 (1), 11–21, 2011.
• [13] M.I. Jinnah and S.C. Mathew, When is the comaximal graph split?, Comm. Algebra, 40 (7), 2400–2404, 2012.
• [14] J. Sedlacek, Some properties of interchange graphs, Theory of Graphs and Its Applications, symposium smolenice, 145–150, 1963.
• [15] K. Samei, On the comaximal graph of a commutative ring, Canad. Math. Bull. 57 (2), 413–423, 2014.
• [16] P.K. Sharma and S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1), 124–127, 1995.
• [17] D. Sinha and A.K. Rao, Co-maximal graph, its planarity and domination number, Journal of Interconnection Networks, 20 (2), 2050005, 2020.
• [18] C. Thomassen, The graph genus problem is NP-complete, Journal of Algorithms, 10 (4), 568–576, 1989.
• [19] S. Visweswaran and J. Parejiya, Some results on the complement of the comaximal ideal graphs of commutative rings, Ric. Mat. 67 (2), 709–728, 2018.
• [20] H.J. Wang, Graphs associated to co-maximal ideals of commutative rings, J. Algebra, 320 (7), 2917–2933, 2008.
• [21] H.J. Wang, Co-maximal graph of non-commutative rings, Linear Algebra Appl. 430 (2-3), 633–641, 2009.
• [22] D. Wang, L. Chen and F. Tian, Automorphisms of the co-maximal ideal graph over matrix ring, J. Algebra Appl. 16 (12), 1750226, 2017.
• [23] A. T. White, Graphs, Groups and Surfaces, second ed., North-Holland Mathematics Studies, 8, North-Holland Publishing Co., Amsterdam, 1984.