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

TL;DR

This paper develops combinatorial methods to count primitive periodic and pseudo orbits on specific circulant graphs, and applies these results to analyze quantum graph properties such as the variance of characteristic polynomial coefficients.

Contribution

It introduces new counting techniques for primitive orbits on circulant graphs and applies them to quantum graph analysis, extending previous methods to more complex orbit structures.

Findings

Counted primitive periodic orbits up to length n on circulant graphs.

Extended counting to primitive pseudo orbits with specific self-intersection properties.

Computed the variance of quantum graph characteristic polynomial coefficients.

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.