On the phase transition phenomenon of graph Laplacian eigenfunctions on trees (with E. Woei), RIMS Kôkyûroku, vol.1743, pp.77-90, 2011.


We discuss our current understanding on the phase transition phenomenon of the graph Laplacian eigenfunctions constructed on a certain type of 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 eigenfunctions 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 can now explain such phase transition phenomenon precisely. For a more complicated class of trees representing neuronal dendrites, we have a conjecture based on the numerical evidence that the number of the eigenvalues larger than 4 is bounded from above by the number of vertices whose degrees is strictly larger than 2. We have also identified a special class of trees that are the only class of trees that can have the exact eigenvalue 4.

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