# Orbits, pseudo orbits, and the characteristic polynomial of q-nary quantum graphs.

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Quantum graphs provide a simple model of quantum mechanics in systems with complex geometry and can be used to study quantum chaos. We evaluate the variance of the coefficients of a quantum binary graph’s associated characteristic polynomial, which is related to the quantum graph’s spectrum. This variance can be written as a finite sum over pairs of short pseudo orbits on the graph with the same topological and metric lengths.To account for all pairs of this type, we first count the numbers of primitive periodic orbits and primitive pseudo orbits on general q-nary graphs by exploiting properties of Lyndon words. We then classify the primitive pseudo orbits on binary graphs by their numbers of self-intersections, the number of repetitions of each self-intersection, and the lengths of self-intersections, in order to determine the contributions of primitive pseudo orbit pairs to the variance. By arranging the sum in a way that considers the contribution of each primitive pseudo orbit paired with all possible partners, we can evaluate the sum over all pairs of primitive pseudo orbits and then use the graph’s ergodicity to asymptotically determine the variance in the limit of large binary graphs. The Bohigas-Giannoni-Schmit conjecture suggests spectral statistics of generic quantum graphs are typically modeled by those of random matrices, in the limit of large graphs. However, we show that, for families of binary graphs, there is a uniform family-specific deviation from random matrix behavior in the variance of coefficients of the characteristic polynomial. Related results for the variance of the coefficients of the characteristic polynomial for general q-nary quantum graphs are also investigated.