GENUS OF COMMUTING GRAPHS OF CERTAIN FINITE GROUPS

Articles in Press

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Cachar College, Silchar-788001, Assam, India.

2 Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India.

Abstract

The commuting graph of a finite group G is a graph whose vertex set is the set of non-central elements of G and two distinct vertices are adjacent if they commute. In this article, we compute genus of commuting graphs of certain classes of finite non-abelian groups and characterize those groups such that their commuting graphs have genus 4, 5 and 6.

Keywords

References

  1. M. Afkhami, M. D. G. Farrokhi and K. Khashyarmanesh, Planar, toroidal, and projective

commuting and noncommuting graphs, Comm. Algebra, 43 (2015), 2964–2970.

  1. A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq., 7

(2000), 139–146.

  1. J. Battle, F. Harary, Y. Kodama, and J. W. T. Youngs, Additivity of the genus of a graph,

Bull. Amer. Math. Soc., 68 (1962), 565–568.

  1. S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math. Mag.,

67 (1994), 366–374.

  1. A. Bouchet, Orientable and nonorientable genus of the complete bipartite graph, J. Com

bin. Theory Ser. B., 24 (1978), 24–33.

  1. R. Brauer and K. A. Fowler, On groups of even order, Ann. Math., 62 (1955), 565–583.
  2. A. K. Das and D. Nongsiang, On the genus of the commuting graphs of finite non-abelian

groups, Int. Electron. J. Algebra, 19 (2016), 91–109.

  1. P. Dutta, B. Bagchi and R. K. Nath, Various energies of commuting graphs of finite

nonabelian groups, Khayyam J. Math., 6 (2020), 27–45.

  1. J. Dutta and R. K. Nath, Finite groups whose commuting graphs are integral, Mat.

Vesnik., 69 (2017), 226–230.

  1. J. Dutta and R. K. Nath, Laplacian and signless Laplacian spectrum of commuting

graphs of finite groups, Khayyam J. Math., 4 (2018), 77–87.

  1. J. Dutta and R. K. Nath, Spectrum of commuting graphs of some classes of finite groups,

Matematika, 33 (2017), 87–95.

  1. P. Dutta and R. K. Nath, Various energies of commuting graphs of some super integral

groups, Indian J. Pure Appl. Math., 52 (2021), 1–10.

  1. W. N. T. Fasfous and R. K. Nath, Inequalities involving energy and Laplacian en

ergy of non-commuting graphs of finite groups, Indian J. Pure Appl. Math., (2023),

https://doi.org/10.1007/s13226-023-00519-7.

  1. W. N. T. Fasfous, R. Sharafdini and R. K. Nath, Common neighborhood spectrum of

commuting graphs of finite groups, Algebra Discrete Math., 32 (2021), 33–48.

  1. D. MacHale, How commutative can a non-commutative group be?, Math. Gaz., 58

(1974), 199–202.

  1. M. Mirzargar and A. R. Ashrafi, Some distance-based topological indices of a non

commuting graph, Hacet. J. Math. Stat., 41 (2012), 515–526.

  1. R. K. Nath, Commutativity degree of a class of finite groups and consequences, Bull.

Aust. Math. Soc., 88 (2013), 448–452.

  1. R. K. Nath, W. N. T. Fasfous, K. C. Das and Y. Shang, Common neighborhood energy

of commuting graphs of finite groups, Symmetry, 13 (2021), 1651 (12 pages).

  1. D. Nongsiang, Double-toroidal and triple-toroidal commuting graph, Hacet. J. Math.

Stat., 53 (2024), 735–747.

  1. D. J. Rusin, What is the probability that two elements of a finite group commute?,

Pacific J. Math., 82 (1979), 237–247, .

  1. R. Sharafdini, R. K. Nath and R. Darbandi, Energy of commuting graph of finite AC

groups, Proyecciones J. Math., 41 (2022), 263–273.

  1. A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, no. 8.,

American Elsevier Publishing Co., Inc., New York, 1973.

Journal of Algebraic Systems

Articles in Press, Accepted Manuscript
Available Online from 09 April 2024

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