New Mathematical Frameworks for Entire Topological Indices and Their Applications in Chemical Modeling

Topological indices are essential for encoding molecular information and have traditionally focused on the properties of individual atoms and their connections. In this work, we study and analyze the first and second entire Zagreb indices. We explore their predictive power through rigorous analysis of octane isomers and benzenoid hydrocarbons, demonstrating their ability to correlate with physical and chemical properties. Furthermore, we derive explicit formulas for these indices for significant families of graphs. To highlight their utility, we apply the first and second entire Zagreb indices in the analysis of drug molecular structures, including graphene, polyomino chains, and benzenoid systems, which are commonly found in molecular drug graphs.