Mysteries around the graph Laplacian eigenvalue 4

TL;DR

This paper investigates the phase transition in eigenvectors of the graph Laplacian on certain trees, revealing a shift from global to localized eigenvectors at eigenvalue 4, with complete analysis for starlike trees and bounds for general graphs.

Contribution

It provides a detailed understanding of the phase transition phenomenon for eigenvectors on specific trees and establishes bounds and decay properties for general graphs.

Findings

Eigenvalue distribution is bell-shaped from 0 to 4 with a jump at 4.

Eigenvectors below 4 are semi-global oscillations, above 4 are localized.

Complete understanding for starlike trees and bounds for general graphs.

Abstract

We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue distribution for such a tree is a smooth bell-shaped curve starting from the eigenvalue 0 up to 4. Then, at the eigenvalue 4, there is a sudden jump. Interestingly, the eigenvectors corresponding to the eigenvalues below 4 are semi-global oscillations (like Fourier modes) over the entire tree or one of the branches; on the other hand, those corresponding to the eigenvalues above 4 are much more localized and concentrated (like wavelets) around junctions/branching vertices. For a special class of trees called starlike trees, we obtain a complete understanding of such phase transition phenomenon. For a general graph, we prove the number of the eigenvalues larger than 4 is bounded from above by the number of vertices whose degrees is strictly higher than 2. Moreover, we also prove that if a graph contains a branching path, then the magnitudes of the components of any eigenvector corresponding to the eigenvalue greater than 4 decay exponentially from the branching vertex toward the leaf of that branch.