  The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights
$\sqrt{d^2_G(u)+d^2_G(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the
degree of the vertex $u$ in $G$. A connected graph $G = (V ,E)$ is called a
quasi-tree, if there exists $u\in V (G)$ such that $G-u$ is a tree. Denote
$\mathscr{Q}(n,k)$=\{$G$: $G$ is a quasi-tree graph of order $n$ with $G-u$
being a tree and $d_G(u)=k$\}. In this paper, we determined the maximum, the
second maximum and the third maximum Sombor indices of all quasi-tree graphs in
$\mathscr{Q}(n,k)$, respectively. Moreover, we characterized their
corresponding extremal graphs, respectively.
