Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor M of the Laplacian of an n-vertex Eulerian digraph Γ gives rise to a finite group Zn−1/Zn−1M known as the sandpile (or critical) group S(Γ) of Γ. We determine S(Γ) of the generalized de Bruijn graphs Γ=DB(n,d) with vertices 0,…,n−1 and arcs (i,di+k) for 0≤i≤n−1 and 0≤k≤d−1, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs.