The Minimal-ABC Trees With ${B}1$ -Branches II
Zhibin Du, Darko Dimitrov · IEEE Access · 2018
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 seeFig. 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 inFig. 1).