An Effective Approach in Fuzzy Graph Molecular Modeling Using Randic Index and Its Applications in Medicinal Chemistry for Parkinson’s Drugs

In graph theory, the Randic index R is a topological graph invariant widely used as a physicochemical descriptor in the mathematical modeling of molecular structures. However, traditional molecular graphs fail to capture the heterogeneity of chemical bonds, since they treat all edges as uniform, ignoring variations in bond lengths and strengths. To overcome this limitation, we adopt a fuzzy graph structure, which provides a more realistic framework for representing molecular interactions. This dual perspective developing rigorous theoretical bounds for the Randic index on fuzzy graph classes and simultaneously applying them to real pharmaceutical compounds ensure both mathematical novelty and practical relevance. We derive analytical bounds for the Randic index over fuzzy versions of complete graphs, paths, stars, and cycles, as well as for graph operations such as Cartesian product, union, join, and composition, confirming the sharpness of the results in each case. To demonstrate the applicability, we compute the Randic index for fuzzy graph representations of Parkinson’s disease drugs (Levodopa, Procyclidine, Trihexyphenidyl, and Apomorphine). The findings indicate that the Randic index, within the fuzzy graph framework, reliably estimates key physicochemical properties such as polarizability, molar refractivity, surface tension, and molar volume, highlighting the strength of combining theoretical results with drug modeling applications.