On the minimal ABC index of trees with k leaves

The atom-bond connectivity (ABC) index of a graph G=(V,E) is defined as ABC(G)=∑uv∈E[d(u)+d(v)−2]/d(u)d(v), where d(u) denotes the degree of vertex u of G. A tree with minimal ABC index among trees with k leaves is said to be k-optimal. In spite of a few attempts, the problem of characterizing k-optimal trees remains open. In the present paper a contracting operation and a splitting operation of a certain graph G that decrease ABC(G) are introduced. With the operations, a few features of k-optimal trees are obtained, which bring us a step closer to the complete solution of the problem.