Clustering of spectra and fractals of regular graphs
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 labelled 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.
Keywords
Cite
@article{arxiv.math/0610742,
title = {Clustering of spectra and fractals of regular graphs},
author = {V. Ejov and J. A. Filar and S. K. Lucas and P. Zograf},
journal= {arXiv preprint arXiv:math/0610742},
year = {2007}
}
Comments
10 pages, 5 figures