Power Graphs as Hypergraphs and <em>n^th</em> Power Graphs as <em>n</em>-SuperHyperGraphs

Graph theory studies the mathematical structures of vertices and edges to model relationships and connectivity [1,2]. A Power Graph has group elements as vertices, with an edge joining two elements whenever one is a power of the other. A Directed Power Graph uses group elements as vertices and places a directed edge x → y whenever y = xm for some m ∈ N. Hypergraphs extend this framework by allowing hyperedges to connect arbitrarily many vertices at once [3], and SuperHyperGraphs further generalize hypergraphs via iterated powerset constructions to capture hierarchical linkages among edges [4,5]. In this paper, we prove that the Power Graph of a group can be realized as a hypergraph and that the Directed Power Graph is a directed hypergraph. Furthermore, we introduce the nth Power Graph and the Directed nth Power Graph, and show that they form subclasses of SuperHyperGraphs and Directed SuperHyperGraphs, respectively.