Degeneracies in the length spectra of metric graphs

Details Degeneracies in the length spectra of metric graphs Degeneracies in the length spectra of metric graphs

2008-07-18

arxiv.org/abs/0807.2903v1

J. Phys. A: Math. Theor. 40 (2007) 10009-10020

Physics

Mathematical Physics

13 pages, 7 figures

Scientific paper

10.1088/1751-8113/40/33/006

The spectral theory of quantum graphs is related via an exact trace formula with the spectrum of the lengths of periodic orbits (cycles) on the graphs. The latter is a degenerate spectrum, and understanding its structure (i.e.,finding out how many different lengths exist for periodic orbits with a given period and the average number of periodic orbits with the same length) is necessary for the systematic study of spectral fluctuations using the trace formula. This is a combinatorial problem which we solve exactly for complete (fully connected) graphs with arbitrary number of vertices.

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