Double-toroidal and triple-toroidal commuting graph

Year 2024, , 735 - 747, 27.06.2024

Deiborlang Nongsiang

https://doi.org/10.15672/hujms.1052964

Abstract

In this paper, all finite non-abelian groups whose commuting graphs can be embed on the double-torus or triple-torus are classified.

Keywords

commuting graph, finite group, genus

References

• [1] A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra 298(2), 468–492, 2006.
• [2] M. Afkhami, M. Farrokhi, K. Khashyarmanesh, Planar, toroidal and projective commuting and non-commuting graphs, Commun. Algebra 43(7), 2964–2970, 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. Akbari, A. Mohammadian, H. Radjavi and P. Raja, On the diameters of commuting graphs, Linear Algebra Appl. 418(1), 161–176, 2006.
• [5] C. Bates, D. Bundy, S. Hart and P. Rowley, A Note on Commuting Graphs for Symmetric Groups, Electron. J. Comb. 16(1), 1–13, 2009.
• [6] J. Battle, F. Harary, Y. Kodama and J. W. T. Youngs, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68, 565–568, 1962.
• [7] R. Brauer and K. A. Fowler, On groups of even order, Ann. Math. 62(3), 565–583, 1955.
• [8] A. K. Das, D. Nongsiang, On the genus of the commuting graphs of finite non-abelian groups, IEJA 19, 91–109, 2016.
• [9] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.6.4, 2013. (http://www.gap-system.org/).
• [10] A. Iranmanesh and A. Jafarzadeh, Characterization of finite groups by their commuting graph, Acta Math. Acad. Paedagog. Nyíregyháziensis 23(1), 7–13, 2007.
• [11] A. R. Moghaddamfar, W. J. Shi, W. Zhou and A. R. Zokayi, On the noncommuting graph associated with a finite group, Sib. Math. J. 46(2), 325–332, 2005.
• [12] A. Mohammadian, On commuting graphs of finite matrix rings, Commun. Algebra 38(3), 988–994, 2010.
• [13] B.H. Neumann, A problem of Paul Erdös on groups, J. Aust. Math. Soc. (Series A) 21(4), 467–472, 1976.
• [14] D. B. West, Introduction to Graph Theory (Second Edition), PHI Learning Private Limited, New Delhi, 2009.
• [15] A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, 8, American Elsevier Publishing Co., Inc., New York, 1973.