The Power Graph as a Hypergraph and Iterated-Powerset Power Graphs as n-SuperHyperGraphs

Graph theory provides a fundamental language for modeling relational structure and connectivity via vertices and edges [1]. For a group 𝐺, the (undirected) power graph has vertex set 𝐺 and joins two distinct elements whenever one is a positive integral power of the other, while the directed power graph orients an arc 𝑥→𝑦 precisely when 𝑦=𝑥^𝑚 for some 𝑚∈ℕ_≥1. Hypergraphs extend ordinary graphs by allowing an edge to connect an arbitrary number of vertices [2], and SuperHyperGraphs further generalize hypergraphs via iterated powerset constructions to capture hierarchical, set-valued incidence patterns. In this paper, we show that the power graph can be viewed as a 2-uniform hypergraph and that the directed power graph admits a natural representation as a directed hypergraph. Motivated by the powerset-based hierarchy of SuperHyperGraphs, we then introduce the 𝑛-level power graph and the directed 𝑛-level power graph, whose vertices lie in iterated nonempty powersets of 𝐺 and whose adjacency is induced by level-wise exponentiation maps. We prove that these constructions fit naturally within the SuperHyperGraph and directed SuperHyperGraph paradigms, respectively.