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

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  • Get the official version via doi:10.1016/j.laa.2012.12.012.

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