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 geometricarithmetic index of a graph G was defined as
GA1 = X
uv∈E
2
√
dudv
du + dv
.
This graph invariant is useful for chemical proposes. The main use of GA1 is for designing so-called quantitative structure-activity relations and quantitative structureproperty relations. In this paper we obtain new inequalities involving the geometricarithmetic 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.We are grateful to the constructive comments from anonymous referee on our pape. The first and third authors are supported by the "Ministerio de Economía y Competititvidad" (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and by the CONACYT (FOMIX-CONACyT-UAGro 249818), Mexico
