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

Abstract

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) groupS(Γ) of Γ. We determine S(Γ) of the generalized de Bruijn graphsΓ=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.Moreover, for a prime p and an n-cycle permutation matrix X∈GLn(p) we show that S(DB(n,p)) is isomorphic to the quotient by 〈X〉 of the centralizer of X in PGLn(p). This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field Fpn from spanning trees in DB(n,p).