On strong nodal domains for eigenfunctions of Hamming graphs

TL;DR

This paper investigates the minimal number of strong nodal domains of eigenfunctions of hypercube Laplacians, confirming a conjecture for many cases and extending results to Hamming graphs with larger alphabets.

Contribution

It confirms a conjecture about eigenfunctions with minimal strong nodal domains for hypercubes and extends the analysis to Hamming graphs with q ≥ 3.

Findings

Confirmed the conjecture for eigenvalues with i ≤ 2/3(n - 1/2) for odd i

Confirmed the conjecture for eigenvalues with i ≤ 2/3(n - 1) for even i

Obtained stronger results for Hamming graphs with q ≥ 3

Abstract

The Laplacian matrix of the $n$-dimensional hypercube has $n+1$ distinct eigenvalues $2i$, where $0\leq i\leq n$. In 2004, B\i y\i ko\u{g}lu, 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 $q\geq 3$.