G-Hypergraphs Induced by Locally Compact Hypergroups

Abstract

In this paper, we introduce the notion of a g-hypergraph as a natural extension and generalization of both classical graph theory and hypergraph theory. The proposed structure provides a broader mathematical framework for studying relationships among elements in algebraic systems, particularly those arising from hypergroups. By combining the structural properties of graphs with the generalized connectivity of hypergraphs, g-hypergraphs offer a flexible and powerful tool for analyzing complex algebraic and topological interactions. We investigate several important classes of g-hypergraphs induced by hypergroups and examine their fundamental structural characteristics.

The paper also examines conditions under which the pair H:=(V,E), consisting of the vertex set and edge set of a g-hypergraph, satisfies desirable topological properties. In particular, we discuss the role of local compactness and provide several applicable sufficient conditions ensuring that the induced g-hypergraph possesses appropriate topological compatibility. These results contribute to the development of a unified framework connecting hypergroup theory, topology, and generalized graph structures. The concepts and results presented in this work may serve as a foundation for further investigations in algebraic graph theory, topological hyperstructures, and related applications in mathematical modeling and discrete structures.