On the maximum ABC index of bipartite graphs without pendent vertices

Abstract For a simple graph G, the atom–bond connectivity index (ABC) of G is defined as ABC(G) = ∑ u v ∈ E ( G ) d ( u ) + d ( v ) − 2 d ( 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, A B C ( G ) ≤ 2 ( n − 6 ) + 2 3 ( 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.