Seidel Laplacian and Seidel Signless Laplacian Energies of Commuting Graph for Dihedral Groups
DOI:
https://doi.org/10.11113/mjfas.v20n3.3465Keywords:
Seidel Laplacian matrix, Seidel signless Laplacian matrix, energy of a graph, commuting graph, dihedral groupsAbstract
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
Issue
Section
License
Copyright (c) 2024 Mamika Ujianita Romdhini
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.