On the maximum ABC index of bipartite graphs without pendent vertices

For a simple graph G, the atom-bond connectivity index (ABC) of G is defined as ABC(G) = Sigma(uv is an element of E(G)) root d(u)+d(v)-2/d(u)d(v), where d(v) denotes the degree of vertex v of G. In this paper, we prove that for any bipartite graph G of order n >= 6, size 2n - 3 with delta(G) >= 2, ABC(G) >= root 2(n - 6) + 2 root 3(n-2)/n-3 + 2, and we characterize all extreme bipartite graphs.