Pseudo A4-graphs and their properties in dihedral groups
Let G be a finite group. We introduce a new graph definition, called the pseudo A4-graph, as an extension of the classical A4-graph. The pseudo A4-graph of G or PA4-graph is a simple graph ΓG whose vertices are elements of G, and two distinct vertices a and b are joined by an edge if and only if ab−1 = ba−1. This paper presents the formulas for certain graph invariants such as the number of edges, diameter, total degree, chromatic number, clique number, independence number, domination number, matching number, and graph energy. The method involves constructing the A4-graphs of dihedral groups, analyzing their structure, and systematically computing graph invariants through algebraic reasoning and combinatorial techniques to establish fundamental properties and relationships. One of the results in this paper comply with the well-known fact that the energy of a graph is always an even integer.