On structural properties of trees with minimal atom-bond connectivity index II: Bounds on B1- and B2-branches

The atom-bond connectivity (ABC) index is a degree-based topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In an earlier paper, it was shown that trees with minimal ABC index do not contain so-called Bk-branches, with k≥5, and that they do not have more than four B4-branches. Our main results here reveal that the number of B1 and B2-branches are also bounded from above by small fixed constants. Namely, we show that trees with minimal ABC index do not contain more than four B1-branches and more than eleven B2-branches.