On the first geometric-arithmetic index of graphs

Let G be a simple connected graph and ^di be the degree of its ith vertex. In a recent paper [D. Vukievi, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 13691376] the "first geometricarithmetic index" of a graph G was defined as G^A1=∑^{di dj}(^di+^dj)2 with summation going over all pairs of adjacent vertices. We obtain lower and upper bounds on G^A1 and characterize graphs for which these bounds are best possible. Moreover, we discuss the effect on G^A1 of inserting an edge into a graph.