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 Z n−1 /Z n−1 M 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.Moreover, for a prime p and an n-cycle permutation matrix X ∈ GL n (p) we show that S(DB(n, p)) is isomorphic to the quotient by X of the centraliser of X in PGL n (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 F p n from spanning trees in DB(n, p).Dedicated to the memory ofÁkos Seress.

The sandpile group of a thick cycle graph

“…Explicit formulae for the sandpile groups of several families of graphs are known. These include complete graphs K n [7,29], complete bipartite graphs K m,n [29], complete multipartite graphs K ⃗ n [25], cycles C n [31], generalized de Bruijn graphs [12], line graphs of graphs [4], Möbius ladders M n [14,20], regular trees [37], threshold graphs [18], square cycles C 2 n [15], twisted bracelets [34], and wheel graphs W n [19]. Sandpile groups of certain Cartesian products of graphs are also known [13,16,24,27,35,38,39].…”