On the maximum ABC index of bipartite graphs without pendent vertices

Abstract

Abstract For a simple graph G , the atom–bond connectivity index ( ABC ) of G is defined as ABC ( G ) = $\sum_{uv\in{}E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$ 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 2 n − 3 with δ ( G ) ≥ 2, $ABC(G)\leq{}\sqrt{2}(n-6)+2\sqrt{\frac{3(n-2)}{n-3}}+2,$ and we characterize all extreme bipartite graphs.