The Euler Sombor index is a recently introduced topological descriptor based on vertex degrees. In this paper, we extend this index to fuzzy graphs by incorporating vertex membership values and edge weights, leading to a more realistic representation of uncertain systems. We derive explicit upper bounds for the fuzzy Euler Sombor index in terms of graph parameters such as the number of vertices, edges, and the maximum membership value, and we establish its behavior under fundamental graph operations, including the Cartesian product, join, and union. To demonstrate its practical relevance, we analyze two complex network models. In a fuzzy medical decision network, the proposed index provides a quantitative ranking of disease classes, yielding a total index value of 28.637 and identifying the most critical conditions for intervention. In a fuzzy communication network, a vulnerability analysis based on node removal reveals that key nodes can cause up to a 46.6% degradation in network integrity. These results show that the fuzzy Euler Sombor index is not only a theoretical extension but also an effective quantitative tool for prioritization, sensitivity analysis, and structural assessment in systems characterized by uncertainty.
Movahedi et al. (Tue,) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: