Hyper-Zagreb index in fuzzy environment and its application

The Zagreb indices (ZIs) are important graph invariants that are used extensively in many different fields in mathematics and chemistry, such as network theory, spectral graph theory, fuzzy graph theory (FGT) and molecular chemistry, etc. The hyper-ZI is introduced especially for fuzzy graphs (FGs) in this study. The study computes this index's bounds for a variety of FG types, including paths, cycles, stars, complete FGs and partial fuzzy subgraphs. It is shown that isomorphic FGs produce the same values for this index. Moreover, interesting connections are established between the hyper-ZI and the second ZI for FGs. Moreover, bounds on this index are found for the following operations: direct product, Cartesian product, composition, join, union, strong product and semi-strong product of two FGs. In the end, the effectiveness of this index is compared with three other topological indices: hyper-ZI for crisp graphs, first ZI for FGs and F-index for FGs, in an analysis of the crime "Murder" in India. While the hyper-ZI for FGs, first ZI for FGs and F-index for FGs yield similar outcomes, the hyper-ZI for FGs demonstrates superior realism in detecting crimes in India compared to its crisp graph counterpart.