The product of two high-frequency Graph Laplacian eigenfunctions is smooth

TL;DR

This paper reveals that on graphs, the product of two high-frequency Laplacian eigenfunctions tends to be smooth, contrasting the continuous case where such products are more oscillatory, highlighting a unique graph property.

Contribution

It identifies and explains a novel phenomenon where high-frequency eigenfunction products on graphs are unexpectedly smooth, differing from continuous analogs.

Findings

High-frequency eigenfunction products are smooth on graphs.

Matching oscillation patterns lead to low Dirichlet energy.

Contrasts with continuous setting where products are more oscillatory.

Abstract

In the continuous setting, we expect the product of two oscillating functions to oscillate even more (generically). On a graph $G=(V,E)$, there are only $|V|$ eigenvectors of the Laplacian $L=D-A$, so one oscillates `the most'. The purpose of this short note is to point out an interesting phenomenon: if $\phi_1, \phi_2$ are delocalized eigenvectors of $L$ corresponding to large eigenvalues, then their (pointwise) product $\phi_1 \cdot \phi_2$ is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.