Analytical Bounds and Applications of the Sombor Index in Fuzzy Graph

Chemical graph theory is an interdisciplinary field that integrates fuzzy graph theory and computational methods to model molecular structures as fuzzy graphs and to address associated mathematical problems. Topological indices assign numerical invariants to network structures; among them, the Sombor index, originally introduced in chemical graph theory, provides a useful measure for quantifying the structure of molecular graphs within a fuzzy graph framework. In this paper, we investigate bounds for the Sombor index across several graph families and operations, including edge addition and deletion, broom graphs, fuzzy star graphs on [Formula: see text] vertices, complete bipartite fuzzy graphs, and the star graph ([Formula: see text]). We also examine the relationship between the Sombor index and various properties of alkanes and octane isomers. Furthermore, we demonstrate a significant correlation between this index and multiple thermodynamic parameters, such as heat of vaporization, entropy, acentric factor, and enthalpy of vaporization, while observing a relatively weak correlation with the heat capacity of octane isomers. Finally, we apply the Sombor index in fuzzy graphs ([Formula: see text]) to identify and rank Indian states according to their crime rates. These results highlight the broad applicability of the Sombor index in both chemical graph theory and real-world network analysis.