Some upper bounds for the atom–bond connectivity index of graphs

AbstractThe 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. Furtula et al. determined the extremal ABC values for chemical trees, and showed that the star tree Sn has the maximal ABC index among all trees. In this work, we show that among all n-vertex graphs with vertex connectivity k, the graph Kk⋁(K1⋃Kn−k−1) 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 β and characterize the unique extremal graph as Kβ⋁Kn−β¯