Clustering of spectra and fractals of regular graphs |
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Authors: | V Ejov SK Lucas |
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Institution: | a School of Mathematics and Statistics, University of South Australia, Mawson Lakes SA 5095, Australia b Department of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807, USA c St. Petersburg Department of Steklov Institute of Mathematics, Fontanka 27, St. Petersburg 191023, Russia |
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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. |
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Keywords: | Regular graph Spectrum Fractal Ihara-Selberg trace formula |
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