  We propose the Kazakov-Migdal model on graphs and show that, when the
parameters of this model are appropriately tuned, the partition function is
represented by the unitary matrix integral of an extended Ihara zeta function,
which has a series expansion by all non-collapsing Wilson loops with their
lengths as weights. The partition function of the model is expressed in two
different ways according to the order of integration. A specific unitary matrix
integral can be performed at any finite $N$ thanks to this duality. We exactly
evaluate the partition function of the parameter-tuned Kazakov-Migdal model on
an arbitrary graph in the large $N$ limit and show that it is expressed by the
infinite product of the Ihara zeta functions of the graph.
