Jon Harrison

Spectral properties of quantum graphs with symmetry

We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs maintain important features of the prototypical quantum star graph model. When the edge lengths respect the cyclic symmetry of the graph the spectrum decomposes into subspectra whose corresponding eigenfunctions transform according to irreducible representations of the cyclic group. We show the subspectra exhibit a new form of intermediate spectral statistics applying techniques developed from star graphs. These are statistics intermediate between Poisson and random matrix statistics. Quantum circulant graphs are one example of a more general class of quantum graphs with symmetry constructed from Cayley graphs of finite groups.