Topological indices and QSPR modeling of gonalgia-associated drug molecules via M-polynomials

Topological indices, derived from graph-theoretical representations of molecular structure, have emerged as powerful tools for predicting the physicochemical properties of chemical compounds. In this study, we investigate a series of fifteen clinically significant drugs associated with the treatment of gonalgia (knee pain). The molecular graphs of these compounds are analyzed using the M-polynomial approach to compute seven key degree-based topological indices: the inverse sum index (ISI), harmonic arithmetic index (HA), inverse symmetric division deg index (ISDD), augmented Zagreb index (AZI), sum-connectivity index (SC), geometric arithmetic index (GA), and sum-Balaban index (SJ). A comprehensive quantitative structure-property relationship (QSPR) analysis is then performed to correlate these indices with critical physicochemical properties, including boiling point (BP), melting point (MP), critical temperature (CT), critical volume (CV), octanol-water partition coefficient (LogP), molar refractivity (MR), and calculated LogP (CLogP). Our results demonstrate strong predictive correlations, with the SC index showing exceptional performance for BP, MP, CT, CV, and MR, while the SJ index was the most effective for predicting LogP and CLogP. Among the regression models tested: linear, polynomial, and logarithmic the quadratic model consistently provided the highest accuracy, highlighting nonlinear relationships between molecular structure and properties. This study confirms that M-polynomial-derived topological indices, combined with polynomial regression, offer a reliable and efficient computational framework for predicting drug-like properties, providing valuable insights for pharmaceutical design and optimization.