On the first geometric–arithmetic index of graphs

Let GG be a simple connected graph and didi be the degree of its iith vertex. In a recent paper [D. Vukičević, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 1369–1376] the “first geometric–arithmetic index” of a graph GG was defined as GA1=∑didj(di+dj)/2 with summation going over all pairs of adjacent vertices. We obtain lower and upper bounds on GA1GA1 and characterize graphs for which these bounds are best possible. Moreover, we discuss the effect on GA1GA1 of inserting an edge into a graph.

► The geometric–arithmetic index (GA)(GA) is a vertex-degree-based graph invariant. ► Lower and upper bounds on GAGA are obtained. ► The change of GAGA upon inserting a new edge into the graph is examined.