Various spectra and energies of commuting graphs of finite rings

Year 2020, Volume: 49 Issue: 6, 1915 - 1925, 08.12.2020

Walaa Nabil Taha Fasfous Rajat Nath , Reza Sharafdini

https://doi.org/10.15672/hujms.540309

Cited By: 7

Abstract

The commuting graph of a non-commutative ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R\setminus Z(R)$ and two vertices $x, y$ are adjacent if and only if $xy = yx$. In this paper, we compute various spectra and energies of commuting graphs of some classes of finite rings and study their consequences.

Keywords

commuting graph, spectrum, energy, finite ring

References

• [1] A. Abdollahi, Commuting graphs of full matrix rings over finite fields, Linear Algebra Appl. 428, 2947–2954, 2008.
• [2] M. Afkhami, Z. Barati, N. Hoseini and K. Khashyarmanesh, A generalization of commuting graphs, Discrete Math. Algorithm. Appl. 7 (1), 1450068 (11 pages), 2015.
• [3] S. Akbari, M. Ghandehari, M. Hadian and A. Mohammadian, On commuting graphs of semisimple rings, Linear Algebra Appl. 390, 345–355, 2004.
• [4] S.M. Buckley, D. Machale, and A.N. Sh´e, Finite rings with many commuting pairs of elements, available at http://archive.maths.nuim.ie/staff/sbuckley/Papers/ bms.pdf.
• [5] J. Dutta and R.K. Nath, Rings having four distinct centralizers, Matrix, M. R. Publications, Assam, 2017, pp. 12–18, Ed. P. Begum.
• [6] P. Dutta and R.K. Nath, A generalization of commuting probability of finite rings, Asian-European J. Math. 11 (2), 1850023 (15 pages), 2018.
• [7] P. Dutta and R.K. Nath, On relative commuting probability of finite rings, Miskolc Math. Notes 20 (1), 225–232, 2019.
• [8] J. Dutta, D.K. Basnet and R.K. Nath, On commuting probability of finite rings, Indag. Math. 28 (2), 272–282, 2017.
• [9] J. Dutta, D.K. Basnet and R.K. Nath, On generalized non-commuting graph of a finite ring, Algebra Colloq. 25 (1), 149–160, 2018.
• [10] J. Dutta, D.K. Basnet and R.K. Nath, A note on n-centralizer finite rings, An. Stiint. Univ. Al. I. Cuza Iasi Math. LXIV (f.1), 161–171, 2018.
• [11] J. Dutta, D.K. Basnet and R.K. Nath, Characterizing some rings of finite order, submitted for publication, available at https://arxiv.org/pdf/1510.08207.pdf.
• [12] J. Dutta, W.N.T. Fasfous and R.K. Nath, Spectrum and genus of commuting graphs of some classes of finite rings, Acta Comment. Univ. Tartu. Math. 23 (1), 5–12, 2019.
• [13] A. Erfanian, K. Khashyarmanesh and Kh. Nafar, Non-commuting graphs of rings, Discrete Math. Algorithm. Appl. 7 (3), 1550027 (7 pages), 2015.
• [14] S.C. Gong, X. Li, G.H. Xu, I. Gutman and B. Furtula, Borderenergetic graphs, MATCH Commun. Math. Comput. Chem. 74, 321–332, 2015.
• [15] I. Gutman, Hyperenergetic molecular graphs, J. Serb. Chem. Soc. 64, 199–205, 1999.
• [16] D. MacHale, Commutativity in finite rings, Amer. Math. Monthly, 83, 30–32, 1976.
• [17] A. Mohammadian, On commuting graphs of finite matrix rings, Comm. Algebra 38, 988–994, 2010.
• [18] R.K. Nath, Various spectra of commuting graphs of n-centralizer finite groups, J. Eng. Science and Tech. 10 (2S), 170–172, 2018.
• [19] R.K. Nath, A note on super integral rings, Bol. Soc. Paran. Mat. 38 (4), 213–218, 2020.
• [20] G.R. Omidi and E. Vatandoost, On the commuting graph of rings, J. Algebra Appl. 10 (3), 521–527, 2011.
• [21] F. Tura, L-borderenergetic graphs, MATCH Commun. Math. Comput. Chem. 77, 37–44, 2017.
• [22] E. Vatandoost and F. Ramezani, On the commuting graph of some non-commutative rings with unity, J. Linear Topological Algebra, 5 (4), 289–294, 2016.
• [23] E. Vatandoost, F. Ramezani and A. Bahraini, On the commuting graph of noncommutative rings of order $p^n q$, J. Linear Topological Algebra, 3 (1), 1–6, 2014.
• [24] H.B. Walikar, H.S. Ramane and P.R. Hampiholi, On the energy of a graph, Graph Connections, Eds. R. Balakrishnan, H.M. Mulder, A. Vijayakumar., pp. 120–123, Allied Publishers, New Delhi, 1999.