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
).