  This paper presents a general existence and uniqueness result for mean field
games equations on graphs ($\mathcal{G}$-MFG). In particular, our setting
allows to take into account congestion effects of almost any form. These
general congestion effects are particularly relevant in graphs in which the
cost to move from one node to another may for instance depend on the proportion
of players in both the source node and the target node. Existence is proved
using a priori estimates and a fixed point argument \`a la Schauder. We propose
a new criterion to ensure uniqueness in the case of Hamiltonian functions with
a complex (non-local) structure. This result generalizes the discrete
counterpart of existing uniqueness results.
