New inequalities involving the Geometric-Arithmetic index

TL;DR

This paper introduces new mathematical inequalities involving the geometric-arithmetic index of graphs, which is useful in chemical graph theory, and characterizes the graphs where these inequalities are tight.

Contribution

It provides novel inequalities for the geometric-arithmetic index, improves existing results, generalizes previous findings, and relates it to other topological indices.

Findings

Derived new inequalities involving GA_1 index

Characterized graphs where inequalities are tight

Connected GA_1 to other topological indices

Abstract

Let $G=(V,E)$ be a simple connected graph and $d_i$ be the degree of its $i$th vertex. In a recent paper [J. Math. Chem. 46 (2009) 1369-1376] the first geometric-arithmetic index of a graph $G$ was defined as $$GA_1=\sum_{ij\in E}\frac{2 \sqrt{d_i d_j}}{d_i + d_j}.$$ This graph invariant is useful for chemical proposes. The main use of $GA_1$ is for designing so-called quantitative structure-activity relations and quantitative structure-property relations. In this paper we obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize the graphs which make the inequalities tight. In particular, we improve some known results, generalize other, and we relate $GA_1$ to other well-known topological indices.