Szeged and Padmakar-Ivan Energies of Non-Commuting Graph for Dihedral Groups
DOI:
https://doi.org/10.11113/mjfas.v20n5.3629Keywords:
Szeged matrix, Padmakar-Ivan matrix, non-commuting graph, dihedral group.Abstract
The graph can represent the molecule structure and the -electron energy derives the concept of graph energy. The graph also can be related to the groups or rings as its vertex set. The non-commutative graph is a type of graph whose construction is determined by the structure of a group. This paper focuses on the energy of the non-commuting graph for dihedral groups using the Szeged and Padmakar-Ivan matrices. Both matrices are constructed based on the distance between two vertices in the graph. The eigenvalues of these matrices lead to the formulation of the graph's energy values. Interestingly, the energies obtained are equal to twice the spectral radius and are hyperenergetic.
References
Abdollahi, A., Akbari, S., & Maimani, H. R. (2006). Non-commuting graph of a group. Journal of Algebra, 298(2), 468–492.
Akram, M., & Naz, S. (2018). Energy of Pythagorean fuzzy graphs with applications. Mathematics, 6(136), 1–27.
Alimon, N. I., Sarmin, N. H., & Erfanian, A. (2023). On the Szeged index and its non-commuting graph. Jurnal Teknologi, 85(3), 105–110.
Angadi, S. A., & Hatture, S. M. (2019). Face recognition through symbolic modelling of face graphs and texture. International Journal of Pattern Recognition and Artificial Intelligence, 33(12).
Bapat, R. B., & Pati, S. (2004). Energy of a graph is never an odd integer. Bulletin of Kerala Mathematics Association, 1, 129–132.
Fath-Tabar, G. H., Doslic, T., & Ashrafi, A. R. (2010). On the Szeged and the Laplacian Szeged spectrum of a graph. Linear Algebra and Its Applications, 443, 662–671.
Gutman, I. (1978). The energy of graph. Ber. Math. Statist. Sekt. Forschungszenturm Graz, 103, 1–22.
Gutman, I., & Dobrynin, A. A. (1998). The Szeged index - a success story. Graph Theory Notes of New York, 34, 37–44.
Habibi, N., & Ashrafi, A. R. (2014). On revised Szeged spectrum of a graph. Tamkang Journal of Mathematics, 45(4), 375–387.
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.
Najdafi-Arani, M. J. (2011). Sharp bounds on the PI and vertex PI energy of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 65, 123–130.
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.
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), 761–779.
Rao, Y., Chen, R., Wu, P., Jiang, H., & Kosari, S. (2021). A survey on domination in vague graphs with application in transferring cancer patients between countries. Mathematics, 9(11), 1258.
Romdhini, M. U. (2023). Spectrum and energy of commuting and non-commuting graphs for dihedral groups (Ph.D. thesis). Universiti Putra Malaysia.
Romdhini, M. U., Nawawi, A., Al-Sharqi, F., & Al-Quran, A. (2024). Characteristic polynomial of power graph for dihedral groups using degree-based matrices. Malaysian Journal of Fundamental and Applied Sciences, 20, 328–335.
Romdhini, M. U., Nawawi, A., & Syechah, B. N. (2024). Seidel Laplacian and Seidel signless Laplacian energies of commuting graph for dihedral groups. Malaysian Journal of Fundamental and Applied Sciences, 20, 701–713.
Romdhini, M. U., Al-Sharqi, F., Nawawi, A., Al-Quran, A., & Rashmanlou, H. (2023). Signless Laplacian energy of interval-valued fuzzy graph and its applications. Sains Malaysiana, 52(07), 2127–2138.
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.
Shi, X., Kosari, S., Talebi, A. A., Sadati, S. H., & Rashmanlou, H. (2022). Investigation of the main energies of picture fuzzy graph and its applications. International Journal of Computational Intelligence Systems, 15(1), 31.
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.
Sun, D., Xu, C., & Zhang, Y. (2016). A novel method of 2D graphical representation for proteins and its application. MATCH Communications in Mathematical and in Computer Chemistry, 75, 431–446.
Xiao, B., Song, Y. Z., & Hall, P. (2011). Learning invariant for object identification by using graph methods. Computer Vision and Image Understanding, 115, 1023–1031.
Zhang, H., Bai, X., Zheng, H., Zhao, H., Zhou, J., Cheng, J., & Lu, H. (2013). Hierarchical remote sensing image analysis via graph Laplacian energy. IEEE Geoscience and Remote Sensing Letters, 10, 396–400.
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