Extremal graphs of GQ Index with given minimum degree and their comparison with ABC and Randi´c Index

Abstract

The Geometric-Quadratic index $(GQ)$ of a graph G is defined as $GQ(G) = {\sum\limits_{\xi \psi \in E(G)} {\sqrt {\frac{{2{d_\xi }{d_\psi }}}{{{d_{\xi}^2} + {d_{\psi}^2 }}}} }}$ where $d_\xi$ represents the degree of vertex $\xi$, is studied within the class of $G(\mu,\mho)$ of simple connected graph with $\mho$ vertices with the minimum of degree $\mu$. The primary theoretical contribution of this article is the characterization of minimizing graphs for vast range of parameters, we provide that when either $\mu$ or even, the extremal graph that minimizes the GQ index is necessarily $\mu$-regular. Moreover, we also determine the extremal graph G that minimize the Geometric-Quadratic value or the lower bound for all $\mu \geq\left\lceil \mu_{0}\right\rceil$, with $\mu_{0}=\varrho_0 (\mho-1)$ and $\varrho_0\approx1$ is found to be the unique positive solution to the equation $(1+\varrho) \sqrt{\varrho^2+1}-2\sqrt{2\varrho}=0$. In such cases where extremal graphs are not explicitly identified, the corresponding lower bound is established. The comparative analysis between the ABC and Randi\'{c} index serves to contextualize the GQ index within the existing molecular descriptors revealing its unique structural sensitivity and validating its potential for application in chemoinformatics. Finally, we derive specialized bounds for the GQ index for chemical trees and molecular graphs.