Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On strong nodal domains for eigenfunctions of Hamming graphs

Alexandr Valyuzhenich, Konstantin Vorob'ev

TL;DR

The paper addresses the problem of minimizing strong nodal domains of eigenfunctions on Hamming graphs, extending prior hypercube results to general Hamming graphs H(n,q) with q≥3. It develops two principled construction schemes based on equitable 2-partitions (type A and type B) and leverages tensor-product composition to produce eigenfunctions with exactly two strong nodal domains for wide ranges of eigenvalues. For q=2, the authors verify the conjecture for i up to roughly 2n/3 (odd/even cases handled separately), while for q≥3 they obtain SND=2 for all i up to n (except the q=3,i=n case), and for i=n with q≥4. These results rely on associated functions from equitable partitions and careful connectivity analyses of positive/negative subgraphs, generalizing the correlation-immunity framework to the nodal-domain context and advancing understanding of spectral structure in Hamming graphs.

Abstract

The Laplacian matrix of the $n$-dimensional hypercube has $n+1$ distinct eigenvalues $2i$, where $0\leq i\leq n$. In 2004, Bıyıkoğlu, Hordijk, Leydold, Pisanski and Stadler initiated the study of eigenfunctions of hypercubes with the minimum number of weak and strong nodal domains. In particular, they proved that for every $1\leq i\leq \frac{n}{2}$ there is an eigenfunction of the hypercube with eigenvalue $2i$ that have exactly two strong nodal domains. Based on computational experiments, they conjectured that the result also holds for all $1\leq i\leq n-2$. In this work, we confirm their conjecture for $i\leq \frac{2}{3}(n-\frac{1}{2})$ if $i$ is odd and for $i\leq \frac{2}{3}(n-1)$ if $i$ is even. We also consider this problem for the Hamming graph $H(n,q)$, $q\geq 3$ (for $q=2$, this graph coincides with the $n$-dimensional hypercube), and obtain even stronger results for all $q\geq 3$.

On strong nodal domains for eigenfunctions of Hamming graphs

TL;DR

The paper addresses the problem of minimizing strong nodal domains of eigenfunctions on Hamming graphs, extending prior hypercube results to general Hamming graphs H(n,q) with q≥3. It develops two principled construction schemes based on equitable 2-partitions (type A and type B) and leverages tensor-product composition to produce eigenfunctions with exactly two strong nodal domains for wide ranges of eigenvalues. For q=2, the authors verify the conjecture for i up to roughly 2n/3 (odd/even cases handled separately), while for q≥3 they obtain SND=2 for all i up to n (except the q=3,i=n case), and for i=n with q≥4. These results rely on associated functions from equitable partitions and careful connectivity analyses of positive/negative subgraphs, generalizing the correlation-immunity framework to the nodal-domain context and advancing understanding of spectral structure in Hamming graphs.

Abstract

The Laplacian matrix of the -dimensional hypercube has distinct eigenvalues , where . In 2004, Bıyıkoğlu, Hordijk, Leydold, Pisanski and Stadler initiated the study of eigenfunctions of hypercubes with the minimum number of weak and strong nodal domains. In particular, they proved that for every there is an eigenfunction of the hypercube with eigenvalue that have exactly two strong nodal domains. Based on computational experiments, they conjectured that the result also holds for all . In this work, we confirm their conjecture for if is odd and for if is even. We also consider this problem for the Hamming graph , (for , this graph coincides with the -dimensional hypercube), and obtain even stronger results for all .

Paper Structure

This paper contains 15 sections, 24 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Preliminaries
  4. Equitable partitions
  5. Equitable partitions: definitions and examples
  6. Multiplication construction
  7. Construction of type A
  8. Construction of type B
  9. Associated functions
  10. Eigenfunctions in $H(2,2)$ and $H(3,3)$
  11. Case $q=2$
  12. Case $q=3$
  13. Case $q\geq 4$
  14. Concluding remarks
  15. Acknowledgements

Key Result

Theorem 1

Let $M$ be a generalized Laplacian of a connected graph with $n$ vertices. Then any eigenfunction $f$ corresponding to the $k$-th eigenvalue of $M$ with multiplicity $r$ has at most $k$ weak nodal domains and $k+r-1$ strong nodal domains:

Figures (4)

  • Figure 1: Partition $(D_1,D_2)$ in $H(3,2)$
  • Figure 2: Functions $\varphi_1$ and $\varphi_2$
  • Figure 3: Functions $\psi_1$, $\psi_2$ and $\psi_3$
  • Figure 4: Eigenfunctions with $3$ and $4$ strong nodal domains in $H(2,3)$ and $H(3,3)$ respectively

Theorems & Definitions (41)

  • Theorem 1: DGLS01, Theorems 1 and 2
  • Theorem 2: BHLPS04, Theorem 3
  • Theorem 3: BHLPS04, Theorem 4
  • Theorem 4: BHLPS04, Theorem 5
  • Conjecture 1: BHLPS04, Conjecture 1
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • proof
  • Corollary 2
  • ...and 31 more