Vertex membership value.

For crisp graphs, the notion of edge geodesic numbers has been known for a long time. But lately, the focus has shifted to investigating this idea in fuzzy graphs, which has resulted in studies of a number of properties. Determining a strong edge geodesic number in the context of a m -polar fuzzy graph (m PFG), where nodes and edges both have m membership values, poses special difficulties that call for creative solutions.Strong geodesic numbers and strong edge geodesic numbers in m PFGs are defined in this study along with a detailed description. It determines an upper bound for strong edge geodesic numbers in a variety of well-known m PFGs. The metric space on the set of all vertices in a graph is associated with the strong edge geodesic distance. This article also discusses the sufficient and required conditions for robust edge geodesic cover. Additionally, isomorphic properties on strong geodesic distance are examined. Along with its various characteristics, the neighborhood notion on strong geodesic distance is also presented. Interesting characteristics of the latter are explored, and the connections between strong geodesic and strong edge geodesic numbers are analyzed. Additionally, the usefulness of strong edge geodesic numbers in m PFGs is illustrated via a practical application.This work expands the scope of fuzzy graph theory and its applications by defining and analyzing these new notions, which offer deeper insights into the structural and dynamic features of m PFGs.长期以来，清晰图（crisp graph）的边测地数概念已为人熟知。但近年来，研究焦点转向了模糊图中的该概念，由此催生了诸多相关性质的研究。在节点与边均带有m个隶属度值的m-极模糊图（m-polar fuzzy graph，mPFG）中求解强边测地数，存在特殊难点，亟需创新性解决方案。本研究首先定义了mPFG中的强测地数与强边测地数，并给出了详尽阐释；同时针对多种经典mPFG，推导得到了其强边测地数的上界。图的全体顶点集所构成的度量空间，与强边测地距离存在关联。本文还探讨了鲁棒边测地覆盖的充分必要条件。此外，本文研究了强测地距离下的同构性质。本文还提出了强测地距离下的邻域概念，并对其多项特性展开探究；同时分析了强测地数与强边测地数之间的关联。此外，本文通过一则实际应用案例，展示了mPFG中强边测地数的应用价值。本研究通过定义与分析这些全新概念，深化了对mPFG结构与动态特性的理解，拓展了模糊图理论及其应用的研究范畴。