Energy of commuting graph of finite AC- groups

Authors

  • Reza Sharafdini Persian Gulf University.
  • Rajat Kanti Nath Tezpur University.
  • Rezvan Darbandi Amirkabir University of Technology.

DOI:

https://doi.org/10.22199/issn.0717-6279-4365

Keywords:

commuting graph, non-commuting graph, AC-group, eigenvalue, energy

Abstract

Let Γ be a graph with the adjacency matrix A. The energy of Γ is the sum of the absolute  values of the eigenvalues of A. In this article we compute the energies of the commuting graphs of some finite groups and discuss some consequences regarding hyperenergetic and borderenergetic graphs.

Author Biographies

Reza Sharafdini, Persian Gulf University.

Department of Mathematics, Faculty of Science.

Rajat Kanti Nath, Tezpur University.

Department of Mathematical Science.

Rezvan Darbandi, Amirkabir University of Technology.

Department of Mathematics and Computer Science.

References

A. Abdollahi, S. Akbari, and H. R. Maimani, “Non-commuting graph of a group”, Journal of algebra, vol. 298, no. 2, pp. 468–492, 2006. https://doi.org/10.1016/j.jalgebra.2006.02.015

A. R. Ashrafi, “On finite groups with a given number of centralizers”, Algebra colloquium, vol. 7, no. 2, pp. 139–146, 2000. https://doi.org/10.1007/s10011-000-0139-5

S. Akbari, A. Mohammadian, H. Radjavi, and P. Raja, “On the diameters of commuting graphs”, Linear algebra and its applications, vol. 418, no. 1, pp. 161–176, 2006. https://doi.org/10.1016/j.laa.2006.01.029

C. Bates, D. Bundy, S. Hart, and P. Rowley, “A note on commuting graphs for symmetric groups”, The electronic journal of combinatorics, vol. 16, no. 1, pp. 1-13, 2009. https://doi.org/10.37236/95

S. M. Belcastro and G. J. Sherman, “Counting centralizers in finite groups”, Mathematics magazine, vol. 67, no. 5, pp. 366–374, 1994. https://doi.org/10.1080/0025570x.1994.11996252

J. Dutta and R. K. Nath, “Spectrum of commuting graphs of some classes of finite groups”, Matematika, vol. 33, no. 1, p. 87, 2017. https://doi.org/10.11113/matematika.v33.n1.812

J. Dutta and R. K. Nath, “Finite groups whose commuting graphs are integral”, Matematicki vesnik, vol. 69, no. 3, pp. 226-230, 2017. [On line]. Available: https://bit.ly/3fSj949

J. Dutta and R. K. Nath, “Laplacian and signless Laplacian spectrum of commuting graphs of finite groups”, Khayyam journal of mathematics, vol. 4, no. 1, pp. 77-87, 2018. https://doi.org/10.22034/KJM.2018.57490

P. Dutta, J. Dutta, and R. K. Nath, “Laplacian spectrum of non-commuting graphs of finite groups”, Indian journal of pure and applied mathematics, vol. 49, no. 2, pp. 205–216, 2018. https://doi.org/10.1007/s13226-018-0263-x

P. Dutta and R. K. Nath, “On Laplacian energy of non-commuting graphs of finite groups”, Journal of linear and topological algebra, vol. 7, no. 2, pp. 121-132, 2018. [On line]. Available: https://bit.ly/3FJLvZb

S. C. Gong, X. Li, G. H. Xu, I. Gutman and B. Furtula, “Borderenergetic graphs”, MATCH communications in mathematical and in computer chemistry, vol. 74, no. 2, pp. 321-332, 2015. [On line]. Available: https://bit.ly/33ziuT1

I. Gutman, “The energy of a graph”, Berichte der Mathematisch-statistischen Sektion im Forschungszentrum Graz, vol. 103, pp. 1-22, 1978.

I. Gutman, “Hyperenergetic molecular graphs”, Journal of the serbian chemical society, vol. 64, pp. 199-205, 1999.

X. Li, Y. Shi, I. Gutman, Graph energy, New York: Springer, 2012.

A. Iranmanesh and A. Jafarzadeh, “Characterization of finite groups by their commuting graph”, Acta mathematica academiae paedagogiace nyíregyháziensis, vol. 23, no. 1, pp. 7-13, 2007. [On line]. Available: https://bit.ly/33DpONi

A. Iranmanesh and A. Jafarzadeh, “On the commuting graph associated with the symmetric and alternating groups”, Journal of algebra and its applications, vol. 7, no. 1, pp. 129–146, 2008. https://doi.org/10.1142/s0219498808002710

G. L. Morgan and C. W. Parker, “The diameter of the commuting graph of a finite group with Trivial Centre”, Journal of algebra, vol. 393, pp. 41–59, 2013. https://doi.org/10.1016/j.jalgebra.2013.06.031

R. K. Nath, “Various spectra of commuting graphs of N-centralizer finite groups”, International Journal of Engineering Science and Technology, vol. 10, no. 2S, pp. 170–172, 2018. https://doi.org/10.21817/ijest/2018/v10i2s/181002s030

B. H. Neumann, “A problem of Paul Erdös on groups”, Journal of the Australian mathematical society, vol. 21, no. 4, pp. 467–472, 1976. https://doi.org/10.1017/s1446788700019303

C. Parker, “The commuting graph of a soluble group”, Bulletin of the London mathematical society, vol. 45, no. 4, pp. 839–848, 2013. https://doi.org/10.1112/blms/bdt005

H. B. Walikar, H. S. Ramane and P. R. Hampiholi, “On the energy of agraph”, in Graph Connections, R. Balakrishnan, H. M. Mulder and A. Vijayakumar, Eds. New Delhi: Allied, 1999, pp. 120-123.

Published

2022-01-28

How to Cite

[1]
R. Sharafdini, R. K. Nath, and R. Darbandi, “Energy of commuting graph of finite AC- groups”, Proyecciones (Antofagasta, On line), vol. 41, no. 1, pp. 263-273, Jan. 2022.

Issue

Vol. 41 No. 1 (2022)

Section

Artículos

Copyright (c) 2022 Reza Sharafdini, Rajat Kanti Nath, Rezvan Darbandi

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

  • No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.