Clustering of spectra and fractals of regular graphs

Ejov, V., Filar, J. A., Lucas, S. K. and Zograf, P. (2007) Clustering of spectra and fractals of regular graphs. Journal of Mathematical Analysis and Applications, 333 236-246. doi:10.1016/j.jmaa.2006.09.072


Author Ejov, V.
Filar, J. A.
Lucas, S. K.
Zograf, P.
Title Clustering of spectra and fractals of regular graphs
Journal name Journal of Mathematical Analysis and Applications   Check publisher's open access policy
ISSN 0022-247X
1096-0813
Publication date 2007-09-01
Sub-type Article (original research)
DOI 10.1016/j.jmaa.2006.09.072
Open Access Status Not yet assessed
Volume 333
Start page 236
End page 246
Total pages 11
Place of publication Maryland Heights, MO, United States
Publisher Academic Press
Language eng
Formatted abstract
We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labeled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara-Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.
Keyword Fractal
Ihara-Selberg trace formula
Regular graph
Spectrum
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
 
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Citation counts: TR Web of Science Citation Count  Cited 7 times in Thomson Reuters Web of Science Article | Citations
Scopus Citation Count Cited 11 times in Scopus Article | Citations
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