| Summary: | Geometric group theory is one of the bridges that connect graph theory and group theory, allowing for the study of one in relation to the other. This research is carried out by defining a graph of group and investigating the systematic relationship among group elements using vertex adjacency of the corresponding defined graph. There are various defined graphs of groups, one of which is an enhanced power graph of group. It is defined as a simple undirected graph where the vertices are all elements from the group and two distinct vertices, x and y are adjacent if they belong to the same cyclic subgroup. The properties of graphs defined on groups can be evaluated through their general presentation. However, selecting a different set of vertices of the defined graph of group will provide us with a different general presentation. Therefore, it is important to focus on choosing another specific set of vertices as a restriction. Let G be a finite group and a new graph known as the deep enhanced power graph of G is defined by considering all the elements except the non-trivial central element of G as its vertices. The deep enhanced power graph is constructed for all finite dihedral groups to determine their patterns and then form the general representation. The classification of the types of graphs is also obtained through the use of the general presentation. The results indicate that the deep enhanced power graph resembles a connected graph with two non-disjoint subgraphs, namely a complete subgraph and a star subgraph. © 2024 Author(s).
|
|---|