Clustering of spectra and fractals of regular graphs

Details Clustering of spectra and fractals of regular graphs Clustering of spectra and fractals of regular graphs

2006-10-25

arxiv.org/abs/math/0610742v3

J. Math. Anal. Appl. 333 (2007) 236-246

Mathematics

Combinatorics

10 pages, 5 figures

Scientific paper

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.

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