Transmission-Based Energies of Prime Coprime Graph for Integers Modulo Group

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.