A Novel Framework of q-Rung Orthopair Fuzzy Ultra-Hypergraphs with Fuzzy Impact Measures

This paper introduces the novel framework of the q-Rung Orthopair Fuzzy Ultra-hypergraph (qROFU), a sophisticated structure that synthesizes the expressive power of q-Rung Orthopair Fuzzy Sets (q-ROFS) with the hierarchical architecture of superhypergraphs. The model is defined by three families of q-ROFSs, representing vertices and hyperedges, with membership and non-membership constraints governed by t-norm and t-conorm operations. We rigorously investigate the fundamental properties of qROFUS through the lens of (a)-cuts, establishing conditions under which the structure reduces to a classical superhypergraph. Furthermore, we define a novel q-Rung Orthopair fuzzy impact measure, E*E*, to quantify the influence and fuzziness within the hypergraph's components. Several key theorems are proven, detailing the hierarchical relationships between the impact measures of the vertex and hyperedge sets, which allows for their classification as weak, strong, and middle. This work provides a comprehensive theoretical foundation for applying advanced fuzzy hypergraphs to model complex systems with multifaceted uncertainty and relational complexity, paving the way for future applications in network analysis and decision-making.

کلیدواژه ها:

q-Rung Orthopair Fuzzy Sets ، q-Rung Orthopair Ultra-Hypergraphs ، Fuzzy Impact ، Complex Network