| Summary: | Enhanced power graphs are important graphs that are extensively studied in characterizing finite groups due to their practical and interesting properties. The properties of the graphs are related to the selection of a set of vertices in their definition, which can be emphasized as one of the developments of the graph. Different sets of vertices selected will lead to the different prop- erties of the graphs. Graph invariants and classification of the graph on groups are essential properties of the defined graph. Graph invariants are the graph properties that remain unchanged after certain transformations, which serve as computational approaches in several areas. Additionally, the classification of the graph is used to establish the defined graphs by the characteristics of the group they represent. A general graph presentation of the desired groups can help researchers to find properties more efficiently. Therefore, this paper highlights the general presentation, some graph invariants, and classification of the deep enhanced power graph of the semi-dihedral groups. The deep enhanced power graph of a finite group G is the graph whose vertices are the elements G except the nontrivial central element, and two different vertices are adjacent if they belong to the same proper cyclic subgroup. It is constructed for all finite semi-dihedral groups to discover their patterns, followed by general presentation formation. Some numerical graph invariants of the deep enhanced power graphs of the semi-dihedral groups are obtained using the general presen- tation: the degree of a vertex, clique number, chromatic number, independence number, domination number, girth, and diameter. For all semi-dihedral groups, the deep enhanced power graph is classified as a connected and perfect graph. © 2024 Author(s).
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