On the phase transition phenomenon of graph Laplacian eigenfunctions on trees (ウェーブレットの発展と理工学的応用--RIMS共同研究報告集)

Content Provider	Semantic Scholar
Author	直樹, 斎藤 Woei, Ernest
Copyright Year	2011
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.
Starting Page	77
Ending Page	90
Page Count	14
File Format	PDF HTM / HTML
Volume Number	1743
Alternate Webpage(s)	https://www.math.ucdavis.edu/~saito/publications/saito_phasetransglapeigtree.pdf
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article