  In 2021, the Sombor index was introduced by Gutman, which is a new
degree-based topological molecular descriptors. The Sombor index of a graph $G$
is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, where
$d_G(v)$ is the degree of the vertex $v$ in $G$. Let $\mathscr{T}_{n,m}$ and
$\mathscr{U}_{n,m}$ be the set of trees and unicyclic graphs on $n$ vertices
with fixed matching number $m$, respectively. In this paper, the tree and the
unicyclic graph with the maximum Sombor index are determined among
$\mathscr{T}_{n,m}$ and $\mathscr{U}_{n,m}$, respectively.
