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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 $sigma_1cdots sigma_n$ is connected to $tau_1cdots tau_n$ 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!

The computation for $k=2$ appears (among other places) on pp. 158--159 of my book Algebraic Combinatorics, second edition. The computation for arbitrary $k$ is completely analogous. Dima Pasechnik is correct that there is one eigenvalue equal to 0, and all the others are equal to $k$.

I believe that for $p$ prime $DB(m,p)$ has one 0 eigenvalue (this is easy to check), and the other eigenvalues are all equal to $p$. This is something that we observed while working on https://arxiv.org/abs/1405.0113, but didn't try to prove (should not be too hard I guess, using a decription of $DB(m,p)$ as the line graph of $DB(m-1,p)$, and then induction).