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

The atom-bond connectivity (ABC) index   is a degree-based topological index, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal ABC index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called BkBk-branches  , with k≥5k≥5, and that they do not have more than four B4B4-branches.