  We study a certain class of classical one dimensional piecewise linear maps.
For these systems we introduce an infinite family of Markov partitions into
equal cells. The symbolic dynamics generated by these systems is described by
bistochastic (doubly stochastic) matrices. We analyze the structure of graphs
generated from the corresponding symbolic dynamics. We demonstrate that the
spectra of quantized graphs corresponding to the regular classical systems have
locally Poissonian statistics, while quantized graphs derived from classically
chaotic systems display statistical properties characteristic of Circular
Unitary Ensemble, even though the corresponding unitary matrices are sparse.

Chaotic Dynamics;Quantum Physics