# Spectral properties of quantum graphs with symmetry.

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Quantum graphs were first introduced as a simple model for studying quantum mechanics in geometrically complex systems. For example, Pauling used quantum graphs to study organic molecules by viewing the atomic nuclei as nodes and the chemical bonds as one-dimensional connecting wires on which the electrons traveled. In 1997, Kottos and Smilansky proposed the use of quantum graphs as a model for studying quantum chaos. Quantum chaos is the study of quantum systems with underlying classical dynamics that exhibit chaos. It is conjectured that the energy levels, or spectra, of quantum systems with classically chaotic dynamics exhibit spacing statistics that are predicted by the Gaussian ensembles of random matrix theory. This contrasts with quantum systems with corresponding integrable classical dynamics which have been shown to be modeled by a Poisson process. However, in certain cases of chaotic dynamics, the spectral statistics fall in a category of intermediate statistics, which combine features associated with the Poisson and random matrix models. In the field of quantum graphs, this is the case with the quantum star graph, one of the simpler models studied. Quantum circulant graphs, the Cayley graphs of cyclic groups, are a natural extension of star graphs because of their rotational symmetry. Chapter three of this dissertation finds secular equations for quantum circulant graphs which are used to study their spectra. In chapter four, the nearest neighbor spacing statistics and two-point correlation function of circulant graphs are numerically analyzed. When the edge lengths of a circulant graph are incommensurate, it displays random matrix statistics; however, when edge length symmetry is introduced, intermediate statistics appear. Predictions for intermediate statistics are also derived analytically and compared to the numerics in this chapter. The quantum graph spectral zeta function for circulant graphs is found in chapter five and used to compute the spectral determinant and vacuum energy. The final chapter examines the spectra of quantum Cayley graphs of general finite groups.