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

The recently introduced atom–bond connectivity (ABCABC) 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 ABCABC values for chemical trees, and showed that the star tree SnSn has the maximal ABCABC index among all trees. In this work, we show that among all nn-vertex graphs with vertex connectivity kk, the graph Kk⋁(K1⋃Kn−k−1)Kk⋁(K1⋃Kn−k−1) is the unique graph with maximum ABCABC index. Furthermore, we determine the maximum ABCABC index of a connected graph with nn vertices and matching number ββ and characterize the unique extremal graph as Kβ⋁Kn−β¯.