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) = ∑uv∈ E(G)d(u)+d(v)−2d(u)d(v), $\sum_{uv\in{}E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$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 δ(G) ≥ 2, ABC(G)≤ 2(n−6)+23(n−2)n−3+2, $ABC(G)\leq{}\sqrt{2}(n-6)+2\sqrt{\frac{3(n-2)}{n-3}}+2,$and we characterize all extreme bipartite graphs.