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

TL;DR

This paper determines the critical groups of generalized de Bruijn and Kautz graphs, extending previous results, and explores their connection to circulant matrices over finite fields, with implications for constructing normal bases.

Contribution

It extends the understanding of critical groups to generalized de Bruijn and Kautz graphs and links these groups to circulant matrices over finite fields.

Findings

Critical groups of generalized de Bruijn and Kautz graphs are explicitly determined.

Critical groups of DB(n,p) relate closely to groups of circulant matrices over _p.

Results suggest methods to construct normal bases in finite fields from spanning trees.

Abstract

We determine the critical groups of the generalized de Bruijn graphs DB$(n,d)$ and generalized Kautz graphs Kautz$(n,d)$, thus extending and completing 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 in close correspondence with groups of $n\times n$ circulant matrices over $\mathbb{F}_p$, which explains numerical data in [OEIS:A027362], and suggests the possibility to construct normal bases in $\mathbb{F}_{p^n}$ from spanning trees in DB$(n,p)$.