  Let $\mathcal{G}(n,k)$ be the set of connected graphs of order $n$ with one
of the Laplacian eigenvalue having multiplicity $k$. It is well known that
$\mathcal{G}(n,n-1)=\{K_n\}$. The graphs of $\mathcal{G}(n,n-2)$ are determined
by Das, and the graphs of $\mathcal{G}(n,n-3)$ with four distinct Laplacian
eigenvalues are determined by Mohammadian et al. In this paper, we determine
the graphs of $\mathcal{G}(n,n-3)$ with three distinct Laplacian eigenvalues,
and then the full characterization of the graphs in $\mathcal{G}(n,n-3)$ is
completed.
