A combinatorial approach to counting primitive periodic and primitive pseudo orbits on circulant graphs

Abstract

For families of 4-regular directed circulant graphs with $n$ vertices, we count the number of primitive periodic orbits of length up to at least $n$. The relevant counting techniques are then extended to count the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) of length up to at least $n$ that lack self-intersections, or that self-intersect only at individual vertices repeated exactly twice (2-encounters of length zero), for two particular families of 4-regular directed circulant graphs. We then regard these two families of graphs as families of quantum graphs and use the counting results to compute the variance of the coefficients of the quantum graph's characteristic polynomial.

arXiv comment36 pages, 2 figures

References in corpus (6)

• Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Periodic-orbit theory of universal level correlations in quantum chaos
• Spectral determinant on quantum graphs
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit