Computations on Valued Fuzzy Ultra-Hypergraphs

This paper introduces valued-fuzzy ultra-hypergraphs, a novel structure synthesizing fuzzy set theory with hierarchical hypergraphs to model complex systems featuring uncertainty, weighted influences, and multi-level relationships. We formalize the structure with fuzzy hypervertices and fuzzy hyperedges, where edge membership is bounded by a function of incident vertex memberships normalized by their weights. Our work establishes a foundational theory, including a complement operation and the concept of isomorphism, with necessary and sufficient conditions for structural equivalence. We further characterize self-complemented hypergraphs and derive the precise relationships required for this property. Additionally, we define the impact membership value of an element, which quantifies its total influence via incident links, and prove this measure is invariant under isomorphism. This framework provides a powerful tool for analyzing real-world networks such as social, biological, and information systems where graded memberships and layered interactions are inherent.

کلیدواژه ها:

Valued-fuzzy ultra-hypergraph ، Isomorphism ، self-complemented ، Impact membership value