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2011.113: Mysteries around the graph Laplacian eigenvalue 4

2011.113: Yuji Nakatsukasa, Naoki Saito and Ernest Woei (2011) Mysteries around the graph Laplacian eigenvalue 4.

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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 transi- tion 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 larger 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. We have also identified a unique class of trees that can have an eigenvalue exactly equal to 4.

Item Type:MIMS Preprint
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2011.113
Deposited By:Yuji Nakatsukasa
Deposited On:19 December 2011

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