On the difference between geometric-arithmetic index and atom-bond connectivity index for trees

Let G be a simple and connected graph with vertex set V(G) and edge set E(G). The geometric-arithmetic index and atom-bond connectivity index of graph G are defined as GA(G) = Sigma(uv is an element of E(G)) 2 root d(u)d(v)/d(u)+d(v) and ABC(G) = Sigma(uv is an element of E(G)) root d(u) + d(v)-2/d(u)d(v), respectively, where the summation extends over all edges uv of G, and d(u) denotes the degree of vertex u in G. Let (GA - ABC) (G) denote the difference between GA and ABC indices of G. In this note, we determine n-vertex binary trees with first three minimum GA - ABC values. We also present a lower bound for GA - ABC index of molecular trees with fixed number of pendant vertices.