2009/08/27 by Idan Oren, Amit Godel, Uzy Smilansky
Mathematics · Physics and Astronomy · #Connection (principal bundle) #Explicit formulae #Geometry #Graph theory and applications #Mathematical analysis #Mathematics #Orbit (dynamics) #Periodic orbits #Pure mathematics #Quantum chaos and dynamical systems #Random walk #Sequence (biology) #Spectral Theory in Mathematical Physics #Statistics #TRACE (psycholinguistics) #math-ph #math.MP #msc:05C80
paper · pdf · doi:10.1088/1751-8113/42/41/415101
22 pages, 3 figures
arxiv created 2009/08/27 · openalex publication_date 2009/09/29 · arxiv updated 2015/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Trace formulae for d -regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w = 1, the only periodic orbits which contribute are the non-back-scattering orbits, and the smooth part in the trace formula coincides with the Kesten–McKay expression. As w deviates from unity, non-vanishing weights are assigned to the periodic walks with backscatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d -regular graphs and the theory of random matrices.