Modeling Hierarchical Systems in Graph Signal Processing, Electric Circuits, and Bond Graphs Via Hypergraphs and Superhypergraphs

Abstract

Graph theory is a core branch of mathematics concerned with representing and analyzing relationships among discrete elements. These concepts are widely used in fields such as electrical engineering. For example, graphs play a crucial role in important frameworks including Graph Signal Processing, Electric Circuits, and Bond Graphs. A hypergraph generalizes the concept of a traditional graph by allowing edges—called hyperedges—to connect more than two vertices simultaneously (Berge, 1984). A superhypergraph further extends this idea by incorporating recursively defined powerset layers, enabling hierarchical and self-referential relationships among hyperedges (Smarandache, 2020). In this paper, we extend the frameworks of Graph Signal Processing, Electric Circuits, and Bond Graphs using hypergraphs and superhypergraphs, and investigate their mathematical properties and illustrative examples. These extensions enable the representation of hierarchical structures inherent in Graph Signal Processing, Electric Circuits, and Bond Graphs, providing a more expressive modeling framework. We anticipate that future research will advance computational experiments and practical applications in these domains.