Mathematical sky: Modeling constellations with hypergraphs

Today, understanding and modeling complex data relationships is critically important in many fields of science and engineering. Across a broad spectrum, from computer science to mathematics, chemistry to social network analysis, structures that can effectively represent connections between data play a key role in finding solutions. This is where the concept of the hypergraph comes into play, as a powerful generalization of traditional graph theory. This thesis comprehensively examines the concept of the hypergraph, which holds significant importance in computer science and mathematics. Starting from the fundamental definitions of hypergraphs, introduced by Claude Berge in 1964, the thesis highlights their differences from normal graph structures and the advantages they offer through an extended range of data. It points out that hypergraphs have been successfully used by generalizing normal graphs in many areas, including database management, data mining, social network analysis, clustering, classification, and prediction. Additionally, it notes that they provide an alternative representation for multilayer networks and are useful when dealing with multi-layered relationships. A significant part of the thesis focuses on the concept of graph energy. It explains how this concept, derived from the spectral properties of a graph, is used in different disciplines like chemistry, physics, and computer science. The thesis states that hypergraph energy is calculated using the eigenvalues of a hypergraph's adjacency matrix, and these energy values are related to the hypergraph's structural properties. It details different hypergraph types (uniform, simple, k-uniform, etc.) and their impact on energy calculations. Furthermore, it emphasizes the strong potential of hypergraphs in modeling complex data relationships, explains their theoretical foundations, and showcases their importance in various application areas, particularly their application in the field of constellations. It focuses on how spectral properties, such as graph energy, provide valuable information for understanding and utilizing hypergraph structures. This work aims to provide both a theoretical foundation and a comprehensive perspective on the practical applications of these structures for researchers, academics, and students interested in hypergraph theory. It's hoped that this thesis will contribute to the understanding of complex systems and inspire future research.