Eigenvectors of the De Bruijn Graph Laplacian: A Natural Basis for the Cut and Cycle Space

TL;DR

This paper explicitly describes the eigenvectors of the De Bruijn graph Laplacian, providing a natural basis for its cut- and cycle-spaces, and reveals a rich algebraic structure in the cycle space across all orders.

Contribution

It derives explicit eigenvectors for the De Bruijn graph Laplacian and introduces a Fourier-like transform, establishing a canonical basis for the graph's cycle and cut spaces.

Findings

Eigenvectors of the De Bruijn Laplacian are explicitly characterized.

A Fourier transform analogue diagonalizes the Laplacian.

The cycle space across all orders forms a graded Hopf algebra.

Abstract

We study the Laplacian of the undirected De Bruijn graph over an alphabet $A$ of order $k$. While the eigenvalues of this Laplacian were found in 1998 by Delorme and Tillich [1], an explicit description of its eigenvectors has remained elusive. In this work, we find these eigenvectors in closed form and show that they yield a natural and canonical basis for the cut- and cycle-spaces of De Bruijn graphs. Remarkably, we find that the cycle basis we construct is a basis for the cycle space of both the undirected and the directed De Bruijn graph. This is done by developing an analogue of the Fourier transform on the De Bruijn graph, which acts to diagonalize the Laplacian. Moreover, we show that the cycle-space of De Bruijn graphs, when considering all possible orders of $k$ simultaneously, contains a rich algebraic structure, that of a graded Hopf algebra.