TY - JOUR

T1 - Clustering of spectra and fractals of regular graphs

AU - Ejov, V.

AU - Filar, J. A.

AU - Lucas, S. K.

AU - Zograf, P.

PY - 2007/9/1

Y1 - 2007/9/1

N2 - 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. © 2006 Elsevier Inc. All rights reserved.

AB - 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. © 2006 Elsevier Inc. All rights reserved.

KW - Fractal

KW - Ihara-Selberg trace formula

KW - Regular graph

KW - Spectrum

UR - http://www.scopus.com/inward/record.url?scp=34248193293&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2006.09.072

DO - 10.1016/j.jmaa.2006.09.072

M3 - Article

AN - SCOPUS:34248193293

VL - 333

SP - 236

EP - 246

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1 SPEC. ISS.

ER -