The minimal-ABC trees with $$B_2$$-branches

Zhibin Du, Darko Dimitrov

Abstract: The atom–bond connectivity (ABC) index is a degree-based graph topological index with a lot of chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. It is known (Chen and Guo in MATCH Commun Math Comput Chem 65:713–722, 2011; Das et al. in MATCH Commun Math Comput Chem 76:159–170, 2011) that among all connected graphs, trees minimize the ABC index (such trees are called minimal-ABC trees). Several structural properties of minimal-ABC trees were proved in the past several years. Here we continue to make a step forward towards the complete characterization of the minimal-ABC trees. In Dimitrov (Discrete Appl Math 204:90–116, 2016), it was shown that a minimal-ABC tree cannot contain more than 11 so-called \(B_2\)-branches. We improve this result by showing that if a minimal-ABC tree of order larger than 39 contains so-called \(B_1\)-branches, then it contains exactly one \(B_2\)-branch, and if a minimal-ABC tree of order larger than 163 contains no \(B_1\)-branch, then it contains at most two \(B_2\)-branches.