New Inequalities Involving the Geometric Arithmetic Index

Let G = (V, E) be a simple connected graph and d(i), be the degree of its ith 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) = Sigma(UV is an element of E) 2 root d(u)d(v)/d(u)+d(v) This graph invariant is useful for chemical proposes. The main use of CA(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, and characterize the graphs which make the inequalities tight. In particular, we improve some known results, generalize other, and we relate to other well-known topological indices.