  We compute spectra of symmetric random matrices defined on graphs exhibiting
a modular structure. Modules are initially introduced as fully connected
sub-units of a graph. By contrast, inter-module connectivity is taken to be
incomplete. Two different types of inter-module connectivity are considered,
one where the number of intermodule connections per-node diverges, and one
where this number remains finite in the infinite module-size limit. In the
first case, results can be understood as a perturbation of a superposition of
semicircular spectral densities one would obtain for uncoupled modules. In the
second case, matters can be more involved, and depend in detail on inter-module
connectivities. For suitable parameters we even find near-triangular shaped
spectral densities, similar to those observed in certain scale-free networks,
in a system of consisting of just two coupled modules. Analytic results are
presented for the infinite module-size limit; they are well corroborated by
numerical simulations.
