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

TL;DR

This paper determines the sandpile groups of generalized de Bruijn and Kautz graphs, extending previous results, and explores their algebraic structure relating to circulant matrices over finite fields, with implications for finite field bases.

Contribution

It provides explicit descriptions of sandpile groups for generalized de Bruijn and Kautz graphs, and links these groups to centralizers in projective linear groups over finite fields.

Findings

Sandpile groups of generalized de Bruijn graphs are explicitly characterized.

Connection established between sandpile groups and centralizers in PGL over finite fields.

Insights offered into constructing normal bases in finite fields from graph spanning trees.

Abstract

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $\Gamma$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(\Gamma)$ of $\Gamma$. We determine $S(\Gamma)$ of the generalized de Bruijn graphs $\Gamma=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, 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 $X\in\mathrm{GL}_n(p)$ we show that $S(\mathrm{DB}(n,p))$ is isomorphic to the quotient by $\langle X\rangle$ of the centralizer of $X$ in $\mathrm{PGL}_n(p)$. 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 $\mathbb{F}_{p^n}$ from spanning trees in $\mathrm{DB}(n,p)$.