A valent-based metric framework for analyzing intuitionistic fuzzy graphs in uncertain environments

This paper introduces a new notion of distance in intuitionistic fuzzy graphs, termed the Valent distance, which incorporates the degrees (valences) of both vertices and edges. Based on this, we define the Valent resolving set Rν and establish the concepts of valent-metric dimension dimν⁡(G) and the valent-edge-metric dimension eIdimν⁡(G). These dimensions extend classical graph invariants by accounting for both structural connectivity and uncertainty inherent in intuitionistic fuzzy environments. The study also defines the Valent edge-resolving set Reν which ensures distinguishability of edges under the Valent distance. Comparative results are presented between classical (crisp) and intuitionistic fuzzy graphs, highlighting cases where the valent-based dimensions provide improved resolution. Applications in soil moisture monitoring, targeted marketing, and healthcare service optimization are discussed to demonstrate the practical significance of the proposed framework. The findings suggest that the valent-edge-metric dimension enhances decision-making in systems characterized by partial truth and incomplete information.