# On the Spectral Radius and Sombor Energy of the Non-Commuting Graph for Dihedral Groups

## DOI:

https://doi.org/10.11113/mjfas.v20n1.3252## Keywords:

Sombor matrix, energy of graph, spectral radius, non-commuting graph, dihedral group## Abstract

The non-commuting graph, denoted by , is defined on a finite group , with its vertices are elements of excluding those in the center of . In this graph, two distinct vertices are adjacent whenever they do not commute in . The graph can be associated with several matrices including the most basic matrix, which is the adjacency matrix, , and a matrix called Sombor matrix, denoted by . The entries of are either the square root of the sum of the squares of degrees of two distinct adjacent vertices, or zero otherwise. Consequently, the adjacency and Sombor energies of is the sum of the absolute eigenvalues of the adjacency and Sombor matrices of , respectively, whereas the spectral radius of is the maximum absolute eigenvalues. Throughout this paper, we find the spectral radius obtained from the spectrum of and the Sombor energy of for dihedral groups of order , where . Moreover, there is an almost linear correlation between the Sombor energy and the adjacency energy of for which is slightly different than reported earlier in previous literature.

## References

Abdollahi, A., Akbari, S. & Maimani, H. R. (2006). Non-commuting graph of a group. Journal of Algebra, 298(2), 468-492.

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.

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

Chattopadhyay, S., Panigrahi, P. & Atik, F. (2018). Spectral radius of power graphs on certain finite groups. Indagationes Mathematicae. 29, 730-737.

Ganie, H. A. & Shang, Y. (2022). On the spectral radius and energy of signless laplacian matrix of digraphs. Heliyon, 8, 1-6.

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

Gowtham, K. J. & Swamy, N. N. (2021). On sombor energy of graphs. Nanosystems: Physics, Chemistry, Mathematics 12(4), 411-417.

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

Gutman, I. (2021). Geometric approach to degree-based topological indices: Sombor indices. MATCH Communications in Mathematical and in Computer Chemistry, 86, 11-16.

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

Khasraw, S. M. S, Ali, I. D. & Haji, R. R. (2020). On the non-commuting graph for dihedral group. Electronic Journal of Graph Theory and Applications, 8(2), 233-239.

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

Lin, Z., Zhou, T. & Miao, L. (2023). On the spectral radius, energy and estrada index of the sombor matrix of graphs. Transactions on Combinatorics, 12(4), 191-205.

Liu, H., You, L., Huang, Y. & Fang, X. (2022). Spectral properties of p-sombor matrices and beyond. MATCH Communications in Mathematical and in Computer Chemistry, 87, 59-87.

Mahmoud, R., Sarmin, N. H. & Erfanian, A. (2017). On the energy of non-commuting graph of dihedral groups. AIP Conference Proceedings, 1830, 1-5.

Neumann, B. H. (1976). A groblem of paul erdos on groups. Journal of the Australian Mathematical Society, 21(4), 467-472.

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. & Shinde, S. S. (2017). Degree Exponent polynomial of graphs obtained by some graph operations. Electronic Notes in Discrete Mathematics, 63, 161-168.

Rather, B.A. & Imran, M. (2022). Sharp Bounds on the Sombor Energy of Graphs. MATCH Communications in Mathematical and in Computer Chemistry, 88, 605–624.

Rather, B. A. & Imran, M. (2023). A note on energy and sombor energy of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 89, 467-477.

Redzepovic, I. & Gutman, I. (2022). Comparing energy and sombor energy – An empirical study. MATCH Communications in Mathematical and in Computer Chemistry, 88, 133-140.

Reja, M. S. & Nayeema, S. M. A. (2023). On sombor index and graph energy. MATCH Communications in Mathematical and in Computer Chemistry, 89, 451-466.

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., Al-Sharqi, F. G., Nawawi, A., Al-Quran, A. & Rashmanlou, H. (2023). Signless Laplacian energy of interval-valued fuzzy graph and its applications. Sains Malaysiana, 52(7), 2127-2137.

Ulker, A., Gursoy, A. & Gursoy, N. K. (2022). The energy and sombor index of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 87, 51-58.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2024 Mamika Ujianita Romdhini, Athirah Nawawi

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