Further results on atom-bond connectivity index of trees

The atom-bond connectivity (ABC) index of a graph GG is defined as ABC(G)=∑uv∈E(G)du+dv−2dudv, where E(G)E(G) is the edge set and dudu is the degree of vertex uu of GG. We give the best upper bound for the ABC index of trees with a perfect matching, and characterize the unique extremal tree, which is a molecular tree. We also give upper bounds for the ABC index of trees with fixed number of vertices and maximum degree, and of molecular trees with fixed numbers of vertices and pendent vertices, and characterize the extremal trees.