On the maximum ABC index of graphs without pendent vertices

Let G be a simple graph. The atom-bond connectivity index (ABC) of G is defined as ABC(G)=∑uv∈E(G)d(u)+d(v)−2d(u)d(v), where d(v) denotes the degree of vertex v of G. We characterize the graphs with n vertices, minimum vertex degree  ≥ 2, and m edges for m=2n−4 and m=2n−3, that have maximum ABC index.