Edge-Dependent Fuzzy Hypergraphs: Mathematical Foundations and Optimization Applications

Classical fuzzy hypergraphs assign one membership value to each vertex, and this value remains the same in all hyperedges containing that vertex. In many applications, however, the role or participation of a vertex may change from one group to another. To represent such situations, we introduce edge-dependent fuzzy hypergraphs, where the membership value of a vertex is specified separately for each incident hyperedge. We define edge-dependent fuzzy subhyperedges, fuzzy [Formula: see text]-subhyperedges, fuzzy subhypergraphs, fuzzy semisubhypergraphs, and fuzzy [Formula: see text]-semisubhypergraphs. We also study fuzzy order, fuzzy size, vertex degree, and weighted vertex degree and obtain some basic results concerning these quantities. Incidence, adjacency, degree, and Laplacian matrices are introduced for the proposed model. Homomorphisms and isomorphisms of edge-dependent fuzzy hypergraphs are defined, and several structural quantities are shown to be invariant under isomorphism. An optimization model for research collaboration networks is included as an application. In this model, the same researcher may have different participation levels in different projects, which illustrates the need for edge-dependent membership in higher-order interactions.