  The statistical mechanics of spin models, such as the Ising or Potts models,
on generic random graphs can be formulated economically by considering the N
--> 1 limit of Hermitian matrix models. In this paper we consider the N --> 1
limit in complex matrix models, which describes vertex models of different
sorts living on random graphs. From the graph theoretic perspective one is
using matrix model and field theory inspired methods to count various classes
of directed graphs.
  We also make some remarks on vertex models on planar random graphs (the N -->
infinity limit) where the resulting matrix models are not generally soluble
using currently known methods. Nonetheless, some particular cases may be mapped
onto known models and hence solved.
