Harmonic-Arithmetic Index: Trees with Maximum Degrees and Comparative Analysis of Antidrugs

Chemical graph theory connects the analysis of molecular structures with mathematical graph theory, allowing for the prediction of chemical and physical properties through the use of topological indices. Among these, the recently introduced Harmonic-Arithmetic (HA) index, proposed by Abeer M. Albalahi et al. in 2023, offers a novel method to quantify molecular and graph structures. It is defined as HA(G)=& sum;mu omega is an element of E(G)4dG(mu)dG(omega)(dG(mu)+dG(omega))2, where dG(mu) and dG(omega) are degrees of nodes mu and omega in G. In this paper, the HA index examines the bounds for a tree T of order n, with a maximum degree triangle. The application of the HA index extends to QSPR/QSAR analyses, where topological indices play a crucial role in predicting the relationship between molecular structures and physicochemical properties, such as in Parkinson's, disease-related antibiotics by calculating their topological indices and analyzing them using QSPR models. Comparative analyses were performed between linear regression models and curvilinear-approach quadratic and cubic regression models to identify the minimal RMSE and enhance predictive accuracy for physicochemical properties. The results demonstrate that the HA index effectively connects mathematical graph theory with molecular characterization, offering reliable predictions, dependable bounds for tree graphs, and meaningful insights into molecular properties. These findings highlight the HA index's potential as a versatile and innovative tool in advancing chemical graph theory and its applications to real-world problems in chemistry.