We quantize graphs (networks) which consist of a finite number of bonds and
vertices. We show that the spectral statistics of fully connected graphs is
well reproduced by random matrix theory. We also define a classical phase space
for the graphs, where the dynamics is mixing and the periodic orbits
proliferate exponentially. An exact trace formula for the quantum spectrum is
developed in terms of the same periodic orbits, and it is used to investigate
the origin of the connection between random matrix theory and the underlying
chaotic classical dynamics. Being an exact theory, and due to its relative
simplicity, it offers new insights into this problem which is at the fore-front
of the research in Quantum Chaos and related fields.Comment: 37 pages, 20 figures, other comments, accepted for publication in the
Annals of Physic