Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups
DOI:
https://doi.org/10.11113/mjfas.v20n3.3422Abstract
This paper introduces cyclic order product prime graphs to examine the spectral graph properties of semi-dihedral groups, specifically focusing on the Laplacian spectrum and energy. Combining principles from cyclic graphs and order product prime graphs enhances understanding of group algebraic structures. In this context, the cyclic order product prime graph for a finite group is defined as one where two distinct vertices are adjacent if their generated subgroup is cyclic, and the product of their orders is a prime power. Our methodological approach begins with establishing a general presentation for these graphs within semi-dihedral groups. This foundational step is essential for deriving some properties, such as vertex degrees, the number of edges, and Laplacian characteristic polynomials. This information subsequently facilitates the determination of the Laplacian spectrum, characterized by seven eigenvalues of various multiplicities, and the computation of their Laplacian energy. The semi-dihedral group of order 16 is a particular example to illustrate the practicality and generality of our theorems.
References
Yang, H., & Hu, G. (2003). Laplacian spectrum analysis and spanning tree algorithm for circuit partitioning problems. Science in China Series F: Information Sciences, 46, 459-465.
Gutman, I., & Zhou, B. (2006). Laplacian energy of a graph. Linear Algebra and Its Applications, 414(1), 29-37.
Ma, X. L., Wei, H. Q., & Zhong, G. (2013). The cyclic graph of a finite group. Algebra, 2013, 1-7.
Aalipour, G., Akbari, S., Cameron, P. J., Nikandish, R., & Shaveisi, F. (2017). On the structure of the power graph and the enhanced power graph of a group. The Electronic Journal of Combinatorics, 24(3), 1-22.
Bello, M., Ali, N. M. M., & Zulkifli, N. (2020). A systematic approach to group properties using its geometric structure. European Journal of Pure and Applied Mathematics, 13(1), 84-95.
Dutta J., and Nath, R. K. (2017). Spectrum of commuting graphs of some classes of finite groups. Matematika, 33(1), 87-95.
Torktaz, M. and Ashrafi, A. (2019). Spectral properties of the commuting graphs of certain groups. AKCE International Journal of Graphs and Combinatorics,16(3), 300-309.
Kumar, J., Dalal, S. and Baghel, V. (2021). On the commuting graph of semidihedral group. Bulletin of the Malaysian Mathematical Sciences Society, 44(5), 3319-3344.
Cheng, T., Dehmer, M., Emmert-Streib, F., Li, Y., & Liu, W. (2021). Properties of commuting graphs over semidihedral groups. Symmetry, 13(1), 103.
Chattopadhyay, S., & Panigrahi, P. (2015). On Laplacian spectrum of power graphs of finite cyclic and dihedral groups. Linear and Multilinear Algebra, 63(7), 1345-1355.
Mehranian, Z., Gholami, A., & Ashrafi, A. R. (2017). The spectra of power graphs of certain finite groups. Linear and Multilinear Algebra, 65(5), 1003-1010.
Hamzeh, A., & Ashrafi, A. R. (2017). Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups. Filomat, 31(16), 5323-5334.
Panda, R. P. (2019). Laplacian spectra of power graphs of certain finite groups. Graphs and Combinatorics, 35(5), 1209-1223.
Parveen, Dalal, S., & Kumar, J. (2023). Enhanced power graphs of certain non-abelian groups. Discrete Mathematics, Algorithms and Applications, 2350063, 1-19.
Mahmoud, R., Sarmin, N. H., & Erfanian, A. (2017). The conjugacy class graph of some finite groups and its energy. Malaysian Journal of Fundamental and Applied Sciences, 13(4), 659-665.
Alimon, N. I., Sarmin, N. H., & Fadzil, A. F. A. (2018). The energy of four graphs of some metacyclic 2-groups. Malaysian Journal of Fundamental and Applied Sciences, 14(1), 59-66.
Sharafdini, R., Nath, R. K., & Darbandi, R. (2022). Energy of commuting graph of finite AC-groups. Proyecciones (Antofagasta), 41(1), 263-273.
Mahmoud, R., Fadzil, A. F. A., Sarmin, N. H., & Erfanian, A. (2019). The Laplacian energy of conjugacy class graph of some finite groups. Matematika, 35(1), 59-65.
Das, K. C., Alazemi, A., & Anđelić, M. (2020). On energy and Laplacian energy of chain graphs. Discrete Applied Mathematics, 284, 391-400.
Sharma, M., & Nath, R. K. (2023). Signless Laplacian energies of non-commuting graphs of finite groups and related results. arXiv preprint arXiv:2303.17795.
Ugasini P. P., Suresh, M., & Bonyah, E. (2023). On the spectrum, energy and Laplacian energy of graphs with self-loops. Heliyon, 9(7), 1-18.
Schaefer, J. and Schlechtweg, K. (2017). On the structure of symmetric spaces of semidihedral groups. Involve, 10(4), 665-676.
Fraleigh, J. B. (2003). A first course in abstract algebra. Pearson Education India.
West, D. B. et al. (2001). Introduction to graph theory. Vol. 2. Prentice Hall Upper Saddle River.
Bello, M. (2021). Order product prime graph and its variations of some finite groups (Doctoral dissertation, Ph. D. Report, Universiti Teknologi Malaysia).
Delen, S., Demirci, M., Cevik, A. S., & Cangul, I. N. (2021). On omega index and average degree of graphs. Journal of Mathematics, 2021, 1-5.
Jahanbani, A., Sheikholeslami, S. M., & Khoeilar, R. (2021). On the spectrum of Laplacian matrix. Mathematical Problems in Engineering, 2021, 1-4.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Norlyda Mohamed, AP Dr Nor Muhainiah, Dr Muhammed Bello
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
