# Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

## DOI:

https://doi.org/10.11113/mjfas.v20n2.3357## Keywords:

Characteristic polynomial, degree-based matrices, power graph, dihedral group## Abstract

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.

## References

Adiga, C. & Smitha, M. (2009). On maximum degree energy of a graph. International Journal of Contemporary Mathematical Sciences, 4(8), 385-396.

Adiga, C. & Swamy, S. (2010). Bounds on the largest of minimum degree eigenvalues of graphs. International Mathematics Forum, 5(37), 1823-1831.

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

Asmarani, E. Y., Syarifudin, A. G., Wardhana, I. G. A. W. & Switrayni, N. W. (2021). The power

graph of dhedral group. Eigen Mathematics Journal, 1(2), 80-85.

Chelvam, T. T. & Sattanathan, M. (2013). Power graph of finite abelian groups. Algebra

and Discrete Mathematics, 16(1), 33-41.

Doley, A., Deka, B., & Bharali, A. (2020). Misbalance degree matrix and related energy of graphs. Journal of Mathematical and Computational Science, 10(3), 436-447.

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

Gutman, I. and Kulli, V. (2021). Nirmala energy. Open Journal of Discrete Applied Mathematics, 4(2), 11-16.

Jafari, S. H. & Zahirofic, S. (2023). Power graphs of all nilpotent groups. Journal of Algebraic Combinatorics, 58, 611-622.

Kumar, A., Selvaganesh, L., Cameron, P. J., & Chelvam, T. (2021). Recent developments on the power graph of finite groups – a survey. AKCE International Journal of Graphs and Combinatorics, 18(2), 65-94.

Ramkumar, R. & Nagarajan, K. (2017). Greatest common divisor degree energy of graphs. International Journal of Mathematical Sciences and Engineering Applications, 11(II), 163-171.

Rad, N., Jahanbani, A., & Gutman, I. (2018). Zagreb energy and zagreb Estrada index of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 79, 371-386.

Ramane, H. S., Gudimani, S. B. & Shinde, S. S. (2013). Signless Laplacian polynomial and characteristic polynomial of a graph. Journal of Discrete Mathematics, 2013, 1-4.

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.

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). Degree sum energy of non-commuting graph for dihedral groups. Malaysian Journal of Science, 41(sp1), 34-39.

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.

Sehgal, A. & Singh, S. N. (2023). The degree of a vertex in the power graph of a finite

Abelian group. Southeast Asian Bulletin of Mathematics, 47, 89-296.

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.

Takshak, N., Sehgal, A., & Malik, A. (2023). Power graph of a finite group is always divisor graph. Asian European Journal of Mathematics, 16(01), 2250236.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2024 Mamika Ujianita Romdhini, Athirah Nawawi, Faisal Al-Sharqi, Ashraf Al-Quran

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