  The $k$-th Laplacian spectral moment of a digraph $G$ is defined as
$\sum_{i=1}^n \lambda_i^k$, where $\lambda_i$ are the eigenvalues of the
Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this
invariant is better known as the Laplacian energy of $G$. We extend recently
published results by characterizing the digraphs which attain the minimal and
maximal Laplacian energy within classes of digraphs with a fixed dichromatic
number. We also determine sharp bounds for the third Laplacian spectral moment
within the special subclass which we define as join digraphs. We leave the full
characterization of the extremal digraphs for $k\ge 3$ as an open problem.
