Energy and Degree Sum Energy of Non-coprime Graphs on Dihedral Groups

Research on graphs has increasingly garnered attention in recent years.This research focuses on graph representations, with particular emphasis on non-coprime graphs within the dihedral group D_{2n} with n = p^k, prime numbers, $k \in \mathbb{Z}^+$. The non-coprime graph of a group G is defined as a graph in which the vertex set is G \{e}, and two distinct vertices r and s are connected by an edge if gcd(|r|,|s|) =\= 1. Specifically, this research examines the adjacency matrix energy and the degree sum energy of non-coprime graphs on dihedral groups. With the extensive application of chemical topological graphs in the field of chemistry, it is hoped that they can assist in the numerical analysis of chemical compounds used in healthcare, such as the analysis of vaccines for the COVID-19 epidemic.