Proof of a conjecture on the geometric-quadratic index of unicyclic graphs and its applications

The geometric-quadratic (GQ) index is defined for a graph $\mathrm{\Gamma }$ as $GQ(\mathrm{\Gamma })=\sum _{{\nu }_i{\nu }_j\in \mathcal{E}(\mathrm{\Gamma })}\sqrt{\frac{2d_i{d}_j}{d_i^2+d_j^2}},$ where $d_i$ denotes the degree of the vertex ${\nu }_i$. This degree-based topological index captures structural information by combining both geometric and quadratic contributions of adjacent vertex degrees. Recently, Furtula and Oz [Geometric-quadratic index from a mathematical perspective, Iranian J. Math. Chem., 16 (2025), 85-89] proposed a conjecture concerning the behavior of the GQ index for unicyclic graphs. In the present paper, we rigorously established the validity of this conjecture, thereby contributing to the theoretical understanding of degree-based graph invariants. Furthermore, Kumar and Das [Comparative study of GQ and QG indices as potentially favorable molecular descriptors, Int. J. Quantum Chem., 124 (2024), #27334] suggested that the GQ index may serve as a more effective molecular descriptor in quantitative structure-property relationship (QSPR) analysis, particularly for predicting physicochemical properties of molecular compounds beyond the extensively studied class of alkane isomers. Motivated by these findings, we further investigated the applicability of the GQ index by examining its role in elucidating QSPR in benzene-based hydrocarbons. For octane isomers, we extended this analysis using linear, quadratic, and cubic regression models across sixteen properties. The cubic model proved most effective for several properties, including four that previously failed under linear models alone. This analysis highlights the broader potential of the GQ index as a chemically meaningful descriptor, including its possible relevance in therapeutic and pharmaceutical contexts.