  The general sum-connectivity index of a graph $G$ is defined as
$\chi_\alpha(G)=\sum\limits_{uv\in E(G)} {(d(u)+d(v))^{\alpha}}$, where $d(v)$
denotes the degree of the vertex $v$ in $G$ and $\alpha$ is a real number. In
this paper it is deduced the maximum value for the general sum-connectivity
index of $n$-vertex trees for $-1.7036\leq \alpha <0$ and of $n$-vertex
unicyclic graphs for $-1\le \alpha <0$ respectively, with fixed maximum degree
$\triangle $. The corresponding extremal graphs, as well as the $n$-vertex
unicyclic graphs with the second maximum general sum-connectivity index for
$n\ge 4$ are characterized. This extends the corresponding results by Du, Zhou
and Trinajsti\' c [arXiv:1210.5043] about sum-connectivity index.
