We will denote by $DB(n,k)$ the directed De Bruijn graph, which is a digraph whose vertices are elements of $\{0,1,\dots,k-1\}^n$, and $\sigma_1\cdots \sigma_n$ is connected to $\tau_1\cdots \tau_{n}$ if and only if $\sigma_i=\tau_{i+1}$ for every $i=1,\dots,n-1$.

We would like to calculate the spectrum of the Laplacian matrix of $DB(n,k)$. Note that since $DB(n,k)$ is $k$-regular, then $\lambda \in Spec(L_{DB(n,k)})$ if and only if $k-\lambda \in Spec(A_{DB(n,k)})$, so we can calculate the eigenvalues of the Laplacian matrix from the eigenvalues of the Adjacency matrix.

We found the following article on the subject, but it only calculates the spectrum of the underlining undirected graph of the De Bruijn graph:

https://core.ac.uk/download/pdf/82810454.pdf

Thanks!