  We present a general method for obtaining the spectra of large graphs with
short cycles using ideas from statistical mechanics of disordered systems. This
approach leads to an algorithm that determines the spectra of graphs up to a
high accuracy. In particular, for (un)directed regular graphs with cycles of
arbitrary length we derive exact and simple equations for the resolvent of the
associated adjacency matrix. Solving these equations we obtain analytical
formulas for the spectra and the boundaries of their support.
