Degeneracies in the length spectra of metric graphs

TL;DR

This paper investigates the degeneracies in the length spectra of metric graphs, providing an exact combinatorial solution for complete graphs to understand the multiplicity of periodic orbit lengths, which is crucial for spectral analysis.

Contribution

It offers an exact combinatorial solution for the degeneracy structure of length spectra in complete graphs, advancing the understanding of spectral fluctuations in quantum graph theory.

Findings

Exact degeneracy counts for complete graphs

Insights into the structure of periodic orbit lengths

Foundation for analyzing spectral fluctuations

Abstract

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.