Characteristics of the Enhanced Power Coprime Graphs of Some Finite Groups

The graphs of groups is a powerful tool for studying finite groups as it visually represents their structures and relationships. In general, introducing new graphs of groups leads to discoveries about the graphs' characteristics, offering significant insights into its structure, connectivity, and spectral aspects. This paper investigates various characteristics of a newly introduced graph called the enhanced power coprime graph of some finite groups. The enhanced power coprime graph of a finite group, G is defined as a graph with elements of G as vertices, where two distinct vertices x and y are adjacent if they generate a proper cyclic subgroup of G and gcd(|x|,|y|)=1. First, we establish the general presentations of the enhanced power coprime graph for all semi-dihedral groups of order 2n and prime power cases of dihedral and generalized quaternion groups. These presentations facilitate the determination of various characteristics, including vertex degrees, clique numbers, chromatic numbers, independence numbers, domination numbers, girth, diameter, graph classification, and the Laplacian spectrum. The results reveal that the enhanced power coprime graphs of the mentioned groups are connected, planar, and perfect, with consistent characteristics for those with similar presentations. These findings have applications in computational group theory, network analysis, and coding theory, using graph characteristics to explore group structures.