Edge 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.