Mysteries around the graph Laplacian eigenvalue 4

We describe our current understanding on the phase transition phenomenon associated with the graph Laplacian eigenvalue λ=4 on trees: eigenvectors for λ<4 oscillate semi-globally while those for λ>4 are concentrated around junctions. For starlike trees, we obtain a complete understanding of this phenomenon. For general graphs, we prove the number of λ>4 is bounded from above by the number of vertices with degrees higher than 2; and if a graph contains a branching path, then the eigencomponents for λ>4 decay exponentially from the branching vertex toward the leaf.