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

The recently introduced atom–bond connectivity ( A B C ) 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 A B C values for chemical trees, and showed that the star tree S n has the maximal A B C index among all trees. In this work, we show that among all n -vertex graphs with vertex connectivity k , the graph K k ⋁ ( K 1 ⋃ K n − k − 1 ) is the unique graph with maximum A B C index. Furthermore, we determine the maximum A B C index of a connected graph with n vertices and matching number β and characterize the unique extremal graph as K β ⋁ K n − β ¯ .