GRAPH ENERGY OF THE COPRIME GRAPH ON GENERALIZED QUATERNION GROUP

This paper investigates the Degree Square Sum Energy , Degree Exponent Energy , and Degree Exponent Sum Energy of the coprime graph on generalized quaternion group ​. This research is quantitative study using previous study as the literature review to construct the new theorem. These energy methods provide new insights into the spectral properties of graphs by their vertex degree distributions into eigenvalue computations. Using spectral graph theory, the general formulas for the , , and of ​ are formulated for for every positive integer . Furthermore, we explore the implications of these methods in understanding the algebraic and spectral characteristics of ​. Numerical results are presented for specific cases to validate the previous theorem. This study contributes to the broader analysis of graph energies, offering a framework for studying other algebraic structures.