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

Anthony Philippakis, Neil Mallinar, Parthe Pandit, Mikhail Belkin

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.

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

Abstract

We study the Laplacian of the undirected De Bruijn graph over an alphabet of order . 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 simultaneously, contains a rich algebraic structure, that of a graded Hopf algebra.

Paper Structure

This paper contains 40 sections, 15 theorems, 93 equations, 4 figures, 1 table.

Table of Contents

  1. Declaration of interest and funding sources:
  2. Introduction
  3. (1) We simultaneously view words of length k and k+1 as the vertices and edges of a db graph, respectively, and as order k and k+1 tensor products of a vector space whose basis is indexed by the letters of our alphabet.
  4. (2) We present a closed-form derivation of the eigenvectors of dB graphs, and we show that they provide natural orthogonal bases for the cut space and cycle space of dB graphs that have not previously been appreciated in the literature.
  5. (3) We demonstrate that, when considering the collection of dB graphs across all possible orders k, the collection of cycle spaces has a rich algebraic structure, that of a graded Hopf algebra.
  6. Notation
  7. Preliminaries
  8. Graph theory
  9. Vector spaces of graphs
  10. De Bruijn graphs
  11. Multilinear algebra
  12. Tensor product on vector spaces
  13. Operators over tensors
  14. Fourier analysis
  15. Discrete Fourier / Hadamard Transform
  16. ...and 25 more sections

Key Result

Theorem 2.1

A directed, connected graph contains an Eulerian cycle if and only if the degree of the in-degree is the same as the out-degree for every vertex in the graph.

Figures (4)

  • Figure 1: The $4^{th}$-order de Bruijn graph, $G_4$, with letters over the alphabet $\mathcal{A} = \{a, b\}$.
  • Figure 2: Circular string composed of the letters $\mathcal{A} = \{a, b\}$.
  • Figure 3: The De Bruijn graph $G_4$ on the alphabet $\mathcal{A} = \{a, b\}$. We highlight two vertices and the directed edge between them. By definition, the incidence matrix encodes $+1$ on the element given by the edge and the incoming vertex, and $-1$ on the element given by the edge and the outgoing vertex. The incidence operator, similarly, acts as $\mathcal{E}(baab) = \theta_L(baab) - \theta_R(baab) = aab - baa$.
  • Figure 4: $\Lambda_V: \mathcal{V}_\mathcal{A}^3 \to \mathcal{V}_\mathcal{A}^3$(left) and $\widehat{\Lambda}_V: \mathcal{V}_\mathcal{A}^3 \to \mathcal{V}_\mathcal{A}^3$(right) for $G_4$, e.g. $\mathcal{A} = \{a, b\}$ with respective Fourier basis given by $\{\bullet, \widehat{b} \}, |\mathcal{A}| = 2, k = 4$. The Laplacian in native space, $\Lambda_V$, has no clear structure, whereas in Fourier space, $\widehat{\Lambda}_V$, it decomposes into block matrices for each Fourier word. We highlight block $n \times n$ tri-diagonal Toeplitz matrices within $\widehat{\Lambda}_V$, with $n=3$ in blue, $n=2$ in red, and $n=1$ in green.

Theorems & Definitions (55)

  • Definition 1: $k$-mers
  • Theorem 2.1: Euler's Theorem
  • Definition 2
  • Definition 3: De Bruijn graph
  • Definition 4: Circular string
  • Example 1: Tensor Product Basis
  • Example 2: Tensor product of vector spaces representing words
  • Definition 5: de Bruijn vector, $\bullet$
  • Example 3: de Bruijn vector, $\bullet$
  • Definition 6: Deletion operators
  • ...and 45 more