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

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.