A combinatorial approach to counting primitive periodic and primitive pseudo orbits on circulant graphs
Abstract
For families of 4regular 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 selfintersections, or that selfintersect only at individual vertices repeated exactly twice (2encounters of length zero), for two particular families of 4regular 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.13051
 Bibcode:
 2021arXiv210713051E
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics;
 05C20;
 05C38;
 81Q35;
 81Q50
 EPrint:
 36 pages, 2 figures