Quantum graphs: a model for quantum chaos

We study the statistical properties of the scattering matrix associated wit h generic quantum graphs. The scattering matrix is the quantum analogue of the classical evolution operator on the graph. For the energy-averaged spec tral form factor of the scattering matrix we have recently derived an exact combinatorial expression. It is based on a sum over families of periodic o rbits which so Far could only be performed in special graphs. Here we prese nt a simple algorithm implementing this summation for any graph. Our result s are in excellent agreement with direct numerical simulations for various graphs. Moreover, we extend our previous notion of an ensemble of graphs by considering ensemble averages over random boundary conditions imposed at t he vertices. We show numerically that the corresponding form factor follows the predictions of random-matrix theory when the number of vertices is lar ge - even when all bond lengths are degenerate. The corresponding combinato rial sum has a structure similar to the one obtained previously by performi ng an energy average under the assumption of incommensurate bond lengths. ( C) 2001 Elsevier Science B.V. All rights reserved.