The Minimal-ABC Trees With ${B}1$ -Branches II

The atom-bond connectivity (ABC) index is one of the most investigated degree-based molecular structure descriptors with a variety of chemical applications. For a given graph $G=(V,E)$ , the ABC index is defined as $ABC(G)=\sum _{uv\in E} {(d_{u}+d_{v}-2)/(d_{u}d_{v})}^{1/2}$ , where $d_{u}$ denotes the degree of the vertex $u$ , and $uv$ is the edge incident to the vertices $u$ and $v$ . It is known that a minimal-ABC tree (a tree with the minimal value of the ABC index) cannot contain more than four so-called $B_{1}$ -branches (the figuration for $B_{1}$ -branch see Fig. 1). Recently, it was shown that a minimal-ABC tree of order larger than 19 contains neither three nor four $B_{1}$ -branches. Here, we further improve those results by showing that a minimal-ABC tree of order larger than 122 cannot contain also one $B_{1}$ -branch. Moreover, we have proven that a minimal-ABC tree of order larger than 122 can contain only two $B_{1}$ -branches and that only in a combination with one $B_{2}$ -branch (the figuration can also see $B_{3}^{**}$ -branch in Fig. 1).