Abstract: Let G=(V,E) be a simple connected graph and di 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 GA1=∑ij∈E2didj−−−√di+dj. This graph invariant is useful for chemical proposes. The main use of GA1 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 GA1 and characterize the graphs which make the inequalities tight. In particular, we improve some known results, generalize other, and we relate GA1 to other well-known topological indices.