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

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 φ 1 , φ 2 are delocalized eigenvectors of L corresponding to large eigenvalues, then their (pointwise) product φ 1 • φ 2 is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.