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For families of 4-regular directed circulant graphs with $n$ vertices, wecount the number of primitive periodic orbits of length up to at least $n$. Therelevant counting techniques are then extended to count the number of primitivepseudo orbits (sets of distinct primitive periodic orbits) of length up to atleast $n$ that lack self-intersections, or that self-intersect only atindividual vertices repeated exactly twice (2-encounters of length zero), fortwo particular families of 4-regular directed circulant graphs. We then regardthese two families of graphs as families of quantum graphs and use the countingresults to compute the variance of the coefficients of the quantum graph'scharacteristic polynomial.

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