General sum-connectivity index of trees with given number of branching vertices

In 2015, Borovi\’{c}anin presented trees with the smallest first Zagreb index among trees with given number of vertices and number of branching vertices. The first Zagreb index is obtained from the general sum-connectivity index if $a = 1$. For $a \in \mathbb{R}$, the general sum-connectivity index of a graph $G$ is defined as $\chi_{a} (G) = \sum_{uv\in E(G)} [d_G (u) + d_G (v)]^{a}$, where $E(G)$ is the edge set of $G$ and $d_G (v)$ is the degree of a vertex $v$ in $G$. We show that the result of Borovi\’{c}anin cannot be generalized for the general sum-connectivity index ($\chi_{a}$ index) if $0 a 1$ or $a 1$. Moreover, the sets of trees having the smallest $\chi_a$ index are not the same for $0 a 1$ and $a 1$. Among trees with given number of vertices and number of branching vertices, we present all the trees with the smallest $\chi_a$ index for $0 a 1$ and $a 1$. Since the hyper-Zagreb index is obtained from the $\chi_a$ index if $a = 2$, results on the hyper-Zagreb index are corollaries of our results on the $\chi_a$ index for $a 1$.