ABC Index of Trees with Fixed Number of Leaves

Given a graph G, the atom-bond connectivity (ABC) index is defined to be ABC(G) = P uv∈E(G) qdG(u)+dG(v)−2 dG(u) dG(v) , where E(G) is the edge set of graph G and dG(v) is the degree of vertex v in graph G. The paper [10] claims to classify tho trees with a fixed number of leaves which minimize the ABC index. Unfortunately, there is a gap in the proof, leading to other examples that contradict the main result of that work. These examples and the problem are discussed in this note