On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. Several structural properties of the trees with minimal ABC index were proved recently, however the complete characterization of the minimal-ABC trees is still an open problem. It is known that minimal-ABC trees can have at most one pendent path of length 3. It is also known that the minimal-ABC trees that have a pendent path of length 3 do not contain so-called Bk-branches, with k ≥ 4, and do not contain more than two B2-branches. Here, we improve the latter result by showing that minimal-ABC trees of order larger than 168 and with a pendent path of length 3 do not contain B2-branches. Moreover, we show that trees with minimal ABC index with a pendent path of length 3 do not contain B1-branches.