Transmission-Based Energies of Prime Coprime Graph for Integers Modulo Group
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Wiener-Hosoya Matrix, Distance Signless Laplacian Matrix, Prime Coprime Graph, Integers Modulo Group, Energy of a Graph
Abstract
Graphs are an excellent instrument that provides an algebraic structure for visualizing and interpreting molecule structures and characteristics. As a result, the problem statement arises regarding how we can interpret graphs with eigenvalues concerning their corresponding matrices. Such questions can be answered by studying spectral graph theory. This research focuses on graphs whose vertex sets are group elements in which the structure of ℤn groups and the definition of a prime coprime graph serve as the foundation for the graph building used in this study. The matrix construction of the graph is based on transmission-based matrices including Weiner-Hosoya and distance signless Laplacian matrices. Research methods include investigating the transmission properties and formulation of the characteristic equation using block matrices. The results obtained are a comprehensive analysis of eigenvalues, spectrum, and spectral radius leading to the prime coprime graph energy for ℤn groups corresponding to both matrices.
References
Akram, M. and S. Naz (2018). Energy of Pythagorean Fuzzy Graphs With Applications. Mathematics, 6(136); 1–27.
Ali, A., I. Gutman, B. Furtula, A. M. Albalahi, and A. E. Hamza (2025). On Chemical and Mathematical Characteristics of Generalized Degree-Based Molecular Descriptors. AIMS Mathematics, 10(3); 6788–6804.
Basak, S. C. (2024). Mathematical Descriptors of Molecules and Biomolecules: Applications in Chemistry, Drug Design, Chemical Toxicology, and Computational Biology. Springer Cham.
Diaz-Diaz, F. and E. Estrada (2025). Signed Graphs in Data Sciences via Communicability Geometry. Information Sciences, 710; 1–20.
Ibrahim, H., R. Sharafdini, T. Reti, and A. Akwu (2021). Wiener-Hosoya Matrix of Connected Graphs. Mathematics, 9(359); 1–12.
Indulal, G., I. Gutman, and A. Vijayakumar (2008). On Distance Energy of Graphs. MATCH Communications in Mathematical and in Computer Chemistry, 60; 461–472.
Khan, S., S. Pirzada, and K. C. Das (2025). On the Sum of Distance Signless Laplacian Eigenvalues of Graphs. Indian Journal of Pure and Applied Mathematics.
Kumar, S., P. Sarkar, and A. Pal (2023). A Study on the Energy of Graphs and Its Applications. Polycyclic Aromatic Compounds, 44(6); 4127–4136.
Masmali, I., M. Azeem, M. K. Jamil, A. Ahmad, and A. N. A. Koam (2024). Study of Some Graph Theoretical Parameters for the Structures of Anticancer Drugs. Scientific Reports, 14; 1–11.
Pirzada, S. and M. O. A. Haq (2023). On the Spread of the Distance Signless Laplacian Matrix of a Graph. Acta Universitatis Sapientiae Informatica, 15(1); 38–45.
Ramane, H. S., G. A. Gudodagi, V. V. Manjalapur, and A. Alhevaz (2019). On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs. Iranian Journal of Mathematical Sciences and Informatics, 14(2); 105–125.
Rana, P., A. Sehgal, P. Bhatia, and P. Kumar (2024). Topological Indices and Structural Properties of Cubic Power Graph of Dihedral Group. Contemporary Mathematics, 5(1); 761–779.
Romdhini, M. U. and A. Nawawi (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.
Sarkar, P., A. Dey, S. Kumar, and A. Pal (2024). On Some Extended Energy of Graphs and Their Applications. Yugoslav Journal of Operations Research, 34(4); 813–824.
Sehgal, A., Manjeet, and D. Singh (2021). Co-Prime Order Graphs of Finite Abelian Groups and Dihedral Groups. Journal of Mathematics and Computer Science, 23(3); 196–202.
Sharafdini, R. and M. Azadimotlagh (2024). A Survey on Automorphism Groups and Transmission-Based Graph Invariants. Journal of Discrete Mathematics and Its Applications, 9(2); 133–146.
Shetty, S. K. and K. A. Bhat (2024). Energy and Spectral Radius of Zagreb Matrix of Graph With Applications. Nanosystems: Physics Chemistry Mathematics, 15(3); 315–324.
Shi, X., S. Kosari, A. A. Talebi, S. H. Sadati, and H. Rashmanlou (2022). Investigation of the Main Energies of Picture Fuzzy Graph and Its Applications. International Journal of Computational Intelligence Systems, 15(1); 31.
Sun, D., C. Xu, and Y. Zhang (2016). A Novel Method of 2D Graphical Representation for Proteins and Its Application. MATCH Communications in Mathematical and in Computer Chemistry, 75; 431–446.
Veeragoudar, J. B., S. V. Patil, and B. S. Bhadre (2022). QSPR Analysis on Graph Energies of Chemical Graphs. Eurasian Chemical Communications, 5; 246–263.
Vijayalakshmi, B., K. Madhumitha, K. Karthiga, and S. Pallavi (2023). Balanced Graph of the Chemical Compound. International Journal of Advances in Engineering and Management, 5(12); 374–377.
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