On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three

The atom-bond connectivity (ABC) index is one of the most investigated degree-based molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length $3$, with the conjecture that it cannot be a case if the order of a tree is larger than $1178$. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length $3$ if its order is larger than $415$.