In 1997 Kottos and Smilansky proposed quantum graphs as a model to study quantum chaos, the quantum mechanics of systems that are classically chaotic. Classical motion on the graph can be defined as a probabilistic Markov chain where the probability of scattering from edge e to edge f is given by the absolute value of the quantum transition amplitude squared, | σ σ e f | 2 . For almost all finite connected quantum graphs the probabilistic dynamics is ergodic and mixing, in other words chaotic.