  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
$$SO_{\alpha}(G)=\sum_{uv\in E} \left[d(u)^2+d(v)^2\right]^\alpha$$ where
$SO(G)=SO_{0.5}(G)$ 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
$\alpha<0$.
