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

TL;DR

This paper investigates the structural properties of trees with minimal atom-bond connectivity (ABC) index, establishing upper bounds on the number of certain branch types, advancing understanding of their optimal configurations.

Contribution

It provides new bounds on the number of B_1 and B_2 branches in trees with minimal ABC index, extending previous structural characterizations.

Findings

Trees with minimal ABC index have at most four B_1-branches.

Trees with minimal ABC index have at most eleven B_2-branches.

Trees with minimal ABC index do not contain B_k-branches for k ≥ 5.

Abstract

The {\em 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~\cite{d-sptmabci-2014}, it was shown that trees with minimal ABC index do not contain so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more than four $B_4$-branches. Our main results here reveal that the number of $B_1$ and $B_2$-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 $B_1$-branches and more than eleven $B_2$-branches.