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

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Abstract

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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