Exploration of Graph Classes and Concepts for SuperHypergraphs and n-th Power Mathematical Structures

Graphs are fundamental concepts used in modeling across various fields such as computer science and engineering. A hypergraph extends this idea by allowing edges, known as hyperedges, to connect any number of vertices. This paper explores superhypergraphs, an advanced extension of hypergraphs that incorporates superedges and supervertices. Examples such as Arboreal Superhypergraphs, Molecular Superhypergraphs, and Probabilistic SuperHyperGraphs demonstrate the diverse structural types that can be modeled using superhypergraphs. Additionally, we introduce the Generalized n-th Powerset, a formalized framework that broadens mathematical applications while retaining the structure of the traditional n-th powerset. By extending the concepts of hypergraphs to superhypergraphs, this work aims to advance both theoretical study and practical applications of these complex structures.