Fuzzy Graph Hyperoperations and Path-Based Algebraic Structures

This paper introduces a framework of hypercompositional algebra on fuzzy graphs by defining and analyzing fuzzy path-based hyperoperations. Building on the notion of strongest strong paths (paths that are both strength-optimal and composed exclusively of strong edges, where each edge achieves maximum connection strength between its endpoints), we define two operations: a vertex-based fuzzy path hyperoperation and an edge-based variant. These operations generalize classical graph hyperoperations to the fuzzy setting while maintaining compatibility with the underlying topology. We prove that the vertex fuzzy path hyperoperation is associative, forming a fuzzy hypersemigroup, and establish additional properties such as reflexivity and monotonicity with respect to α -cuts. Structural features such as fuzzy strong cut vertices and edges are examined, and a fuzzy distance function is introduced to quantify directional connectivity strength. We define an equivalence relation based on mutual full-strength reachability and construct a quotient fuzzy graph that reflects maximal closed substructures under the vertex fuzzy path hyperoperation. Applications are discussed in domains such as trust networks, biological systems, and uncertainty-aware communications. This work aims to lay the algebraic foundations for further exploration of fuzzy hyperstructures that support modeling, analysis, and decision-making in systems governed by partial and asymmetric relationships.