Seidel Laplacian and Seidel Signless Laplacian Energies of Commuting Graph for Dihedral Groups

Authors

  • Mamika Ujianita Romdhini Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Mataram, Mataram 83125, Indonesia
  • Athirah Nawawi Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.
  • Bulqis Nebula Syechah Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Mataram, Mataram 83125, Indonesia

DOI:

https://doi.org/10.11113/mjfas.v20n3.3465

Keywords:

Seidel Laplacian matrix, Seidel signless Laplacian matrix, energy of a graph, commuting graph, dihedral groups

Abstract

In this paper, we discuss the energy of the commuting graph. The vertex set of the graph is dihedral groups and the edges between two distinct vertices represent the commutativity of the group elements. The spectrum of the graph is associated with the Seidel Laplacian and Seidel signless Laplacian matrices. The results are similar to the well-known fact that the energies are not odd integers. We also highlight the relation that the Seidel signless Laplacian energy is never less than Seidel Laplacian energy. Ultimately, we classify the graphs according to the energy value as the hyperenergetic.

References

Aschbacher, M. (2000). Finite group theory. Cambridge, UK: University Press.

Bapat, R.B. & Pati, S. (2004). Energy of a graph is never an odd integer. Bulletin of Kerala. Mathematics Association, 1, 129-132.

Brauer, R. & Fowler, K. A. (1955.) On groups of even order. Annals of Mathematics, 62(3), 565-583.

Brouwer, A. E. & Haemers W. H. (2011). Spectra of graphs. New York: Springer-Verlag.

Gantmacher, F. R. (1959). The theory of matrices. New York: Chelsea Publishing Company.

Gutman, I. (1978). The energy of graph. Ber. Math. Statist. Sekt. Forschungszenturm Graz, 103, 1-22.

Horn, R. A. & Johnson, C. A. (1985). Matrix analysis. Cambridge, UK: Cambridge University Press.

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

Pirzada, S. & Gutman, I. (2008). Energy of a graph is never the square root of an odd integer. Applicable Analysis and Discrete Mathematics, 2, 118-121.

Ramane, H. S., Jummannaver, R. B. & Gutman, I. (2017). Seidel Laplacian energy of graphs. International Journal of Applied Graph Theory, 1(2), 74-82.

Ramane, H. S., Gutman, I., Patil, J. B. & Jummannaver, R. B. (2017). Seidel signless Laplacian energy of graphs. Mathematics Interdisciplinary Research, 2, 181-191.

Rana, P., Sehgal, A., Bhatia, P., Kumar, P. 2024. Topological indices and structural properties of cubic power graph of dihedral group. Contemporary Mathematics, 5(1), 761-79.

Romdhini, M. U., Nawawi, A. & Chen, C. Y. (2022). Degree exponent sum energy of commuting graph for dihedral groups. Malaysian Journal of Science, 41(sp1), 40-46.

Romdhini, M. U. & Nawawi, A. (2022). Maximum and minimum degree energy of commuting graph for dihedral groups. Sains Malaysiana, 51(12), 4145-4151.

Romdhini, M. U. & Nawawi, A. (2023). Degree subtraction energy of commuting and non-commuting graphs for dihedral groups. International Journal of Mathematics and Computer Science, 18(3), 497-508.

Romdhini, M. U., Nawawi, A. & Chen, C. Y. (2023). Neighbors degree sum energy of commuting and non-commuting graphs for dihedral groups. Malaysian Journal of Mathematical Sciences, 17(1), 53-65.

Romdhini, M. U. & Nawawi, A. (2024). On the spectral radius and sombor energy of the non-commuting graph for dihedral groups. Malaysian Journal of Fundamental and Applied Sciences, 20, 65-73.

Sehgal, A., Manjeet, Singh, D. (2021). Co-prime order graphs of finite Abelian groups and dihedral groups. Journal of Mathematics and Computer Science, 23(3), 196-202.

Van Lint, J. H. and J. J. Seidel, J. J. (1966). Equilateral point sets in elliptic geometry. Indagationes Mathematicae, 28, 335-348.

Downloads

Published

26-06-2024

Issue

Section

Article