Summary: The recently introduced atom-bond connectivity (ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. B. Furtula, A. Graovac and D. Vukičević [Discrete Appl. Math. 157, No. 13, 2828–2835 (2009; Zbl 1209.05252)] determined the extremal ABC values for chemical trees, and showed that the star tree has the maximal ABC index among all trees.
In this work, we show that among all n-vertex graphs with vertex connectivity , the graph is the unique graph with maximum ABC index. Furthermore, we determine the maximum ABC index of a connected graph with n vertices and matching number and characterize the unique extremal graph as .