  We define a class of quantum systems called regular quantum graphs. Although
their dynamics is chaotic in the classical limit with positive topological
entropy, the spectrum of regular quantum graphs is explicitly computable
analytically and exactly, state by state, by means of periodic orbit
expansions. We prove analytically that the periodic orbit series exist and
converge to the correct spectral eigenvalues. We investigate the convergence
properties of the periodic orbit series and prove rigorously that both
conditionally convergent and absolutely convergent cases can be found. We
compare the periodic orbit expansion technique with Lagrange's inversion
formula. While both methods work and yield exact results, the periodic orbit
expansion technique has conceptual value since all the terms in the expansion
have direct physical meaning and higher order corrections are obtained
according to physically obvious rules. In addition our periodic orbit
expansions provide explicit analytical solutions for many classic text-book
examples of quantum mechanics that previously could only be solved using
graphical or numerical techniques.
