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

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 $ϕ_1, ϕ_2$ are delocalized eigenvectors of $L$ corresponding to large eigenvalues, then their (pointwise) product $ϕ_1 \cdot ϕ_2$ is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.

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