On structural properties of trees with minimal atom-bond connectivity index II

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications. The problem of complete characterization of trees with minimal ABC index is still an open problem. In [14], 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.