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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. Inthis paper it is deduced the maximum value for the general sum-connectivityindex of $n$-vertex trees for $-1.7036\leq \alpha <0$ and of $n$-vertexunicyclic graphs for $-1\le \alpha <0$ respectively, with fixed maximum degree$\triangle $. The corresponding extremal graphs, as well as the $n$-vertexunicyclic graphs with the second maximum general sum-connectivity index for$n\ge 4$ are characterized. This extends the corresponding results by Du, Zhouand Trinajsti\' c [arXiv:1210.5043] about sum-connectivity index.

General sum-connectivity index of trees and unicyclic graphs with fixed maximum degree