Mysteries around graph Laplacian eigenvalue 4 (with Y. Nakatsukasa and E. Woei), Linear Algebra and its Applications, vol.438, no.8, pp.3231-3246, 2013.
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.
Keywords: graph Laplacian; localization of eigenvectors; phase transition phenomena; starlike trees; dendritic trees; Gerschgorin's disks
Get the full paper: PDF file (revised and corrected on 08/22/12).
Get the official version via doi:10.1016/j.laa.2012.12.012.
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