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We determine the critical groups of the generalized de Bruijn graphsDB$(n,d)$ and generalized Kautz graphs Kautz$(n,d)$, thus extending andcompleting earlier results for the classical de Bruijn and Kautz graphs.Moreover, for a prime $p$ the critical groups of DB$(n,p)$ are shown to be inclose correspondence with groups of $n\times n$ circulant matrices over$\mathbb{F}_p$, which explains numerical data in [OEIS:A027362], and suggeststhe possibility to construct normal bases in $\mathbb{F}_{p^n}$ from spanningtrees in DB$(n,p)$.

Critical groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields: an extended abstract