Note on Sombor index of connected graphs with given degree sequence

For a simple connected graph $G=(V,E)$, let $d(u)$ be the degree of the vertex $u$ of $G$. The general Sombor index of $G$ is defined as $$S{O}_{\alpha }(G)=\sum _{uv\in E}{[d(u{)}^2+d(v{)}^2]}^{\alpha }$$ where $SO(G)=S{O}_{0.5}(G)$ is the recently invented Sombor index. In this paper, we show that in the class of connected graphs with a fixed degree sequence (for which the minimum degree being equal to one), there exists a special extremal $BFS$-graph with minimum general Sombor index for $0<\alpha <1$ (resp. maximum general Sombor index for either $\alpha >1$ or $\alpha <0$). Moreover, for any given tree, unicyclic, and bicyclic degree sequences with minimum degree 1, there exists a unique extremal $BFS$-graph with minimum general Sombor index for $0<\alpha <1$ and maximum general Sombor index for either $\alpha >1$ or $\alpha <0$.