Characteristic Polynomial and Energy of Matrices Associated with Power Graphs of Cyclic Groups
Abstract
Algebraic graph theory is a fast-expanding field with numerous applications, particularly in the study of power graphs of finite groups. This study focuses on the power graph of cyclic groups with prime power order. Several key results are presented, including the characteristic polynomial and the energies of different matrices associated with the power graph. These matrices include the degree sum, degree exponent, degree subtraction, maximum degree, and degree square sum matrix. The research establishes characteristic polynomial for each matrix and computes their respective energies, revealing the complex relationships between the structure of the graph and its algebraic properties. This work extends the understanding of power graphs and contributes valuable insights into their algebraic and spectral properties.
Full text article
References
R. Balakrishnan, “The energy of a graph,” Linear Algebra and its Applications, vol. 387, pp. 287–295, 2004. https://doi.org/10.1016/j.laa.2004.02.038.
N. Nurhabibah, I. G. A. W. Wardhana, and N. W. Switrayni, “Numerical invariants of coprime graph of a generalized quaternion group,” Journal of the Indonesian Mathematical Society, vol. 29, no. 1, pp. 36–44, 2023. https://doi.org/10.22342/jims.29.1.1245.36-44.
L. R. W. Putra, Z. Y. Awanis, S. Salwa, Q. Aini, and I. G. A. W. Wardhana, “The power graph representation for integer modulo group with power prime order,” BAREKENG: Journal of Mathematics and its Applications, vol. 17, no. 3, pp. 1393–1400, 2023. https://doi.org/10.30598/barekengvol17iss3pp1393-1400.
S. T. Lestari, J. R. Albaracin, and I. G. A. W. Wardhana, “Certain indexes of unit graph in integer modulo rings with specific orders,” BAREKENG: Journal of Mathematics and its Applications, vol. 19, no. 4, pp. 2455–2466, 2025. https://doi.org/10.30598/barekengvol19iss4pp2455-2466.
P. J. Cameron and S. Ghosh, “The power graph of a finite group,” Discrete Mathematics, vol. 311, no. 13, pp. 1220–1222, 2011. https://doi.org/10.1016/j.disc.2010.02.011.
S. Bera, “On the intersection power graph of a finite group.,” Electron. J. Graph TheoryAppl., vol. 6, no. 1, pp. 178–189, 2018. https://dx.doi.org/10.5614/ejgta.2018.6.1.13.
L. A. Dupont, D. G. Mendoza, and M. Rodriguez, “The rainbow connection number of the enhanced power graph of a finite group.,” Electronic Journal of Graph Theory & Applications, vol. 11, no. 1, 2023. https://dx.doi.org/10.5614/ejgta.2023.11.1.19.
A. R. Ashrafi, A. Gholami, and Z. Mehranian, “Automorphism group of certain power graphs of finite groups.,” Electron. J. Graph Theory Appl., vol. 5, no. 1, pp. 70–82, 2017. https://dx.doi.org/10.5614/ejgta.2017.5.1.8.
G. Y. Karang, I. G. A. W. Wardhana, and N. H. Sarmin, “Energy and degree sum energy of non-coprime graphs on dihedral groups,” Journal of The Indonesian Mathematical Society, vol. 31, no. 1, p. 1900, 2025. https://doi.org/10.22342/jims.v31i1.1900.
G. Y. Karang, I. G. A. W. Wardhana, and M. Angamuthu, “Energy of non-coprime graph on modulo group,” BAREKENG: Journal of Mathematics and its Applications, vol. 19, no. 4, pp. 2937–2952, 2025. https://doi.org/10.30598/barekengvol19iss4pp2937-2952.
L. R. W. Putra, J. R. Albaracin, and I. G. A. W. Wardhana, “On sombor energy of the nilpotent graph of the ring of integers modulo ε,” Journal of the Indonesian Mathematical Society, vol. 31, no. 3, pp. 1856–1856, 2025. https://doi.org/10.22342/jims.v31i3.1856
S. Hande, S. Jog, and D. Revankar, “Bounds for the degree sum eigenvalue and degree sum energy of a common neighborhood graph,” International Journal of Graph Theory, vol. 1, no. 4, pp. 131–136, 2013.
S. M. Hosamani and H. S. Ramane, “On degree sum energy of a graph,” European Journal of Pure and Applied Mathematics, vol. 9, no. 3, pp. 340–345, 2016. https://www.ejpam.com/ejpam/article/view/2419
H. S. Ramane and S. S. Shinde, “Degree exponent polynomial and degree exponent energy of graphs,” Indian J. Discrete Math, vol. 2, no. 1, pp. 1–7, 2016. https://www.researchgate.net/publication/338195221_DEGREE_EXPONENT_POLYNOMIAL_AND_DEGREE_EXPONENT_ENERGY_OF_GRAPHS
H. S. Ramane and H. N. Maraddi, “Degree subtraction adjacency eigenvalues and energy of graphs obtained from regular graphs,” Open J. Discret. Appl. Math, vol. 1, pp. 8–15, 2018. https://www.doi.org/10.30538/psrp-odam2018.0002.
C. Adiga and M. Smitha, “On maximum degree energy of a graph,” Int. J. Contemp. Math. Sci, vol. 4, no. 8, pp. 385–396, 2009. https://m-hikari.com/ijcms-password2009/5-8-2009/smithaIJCMS5-8-2009.pdf
B. Basavanagoud and E. Chitra, “Degree square sum energy of graphs,” Int. J. Math. And Appl., vol. 55, p. 7, 2018.
K. J. Gowtham and S. N. Narasimha, “On sombor energy of graphs,” Наносистемы: физика, химия, математика, vol. 12, no. 4, pp. 411–417, 2021. https://doi.org/10.17586/2220-8054-2021-12-4-411-417.
I. Gutman, B. Furtula, and S¸. B. Bozkurt, “On randi´c energy,” Linear Algebra and its Applications, vol. 442, pp. 50–57, 2014. https://doi.org/10.1016/j.laa.2013.06.010.
Authors
Copyright (c) 2026 Journal of the Indonesian Mathematical Society
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Article Details
