  We consider two-dimensional N=(2,2) supersymmetric gauge theory on
discretized Riemann surfaces. We find that the discretized theory can be
efficiently described by using graph theory, where the bosonic and fermionic
fields are regarded as vectors on a graph and its dual. We first analyze the
Abelian theory and identify its spectrum in terms of graph theory. In
particular, we show that the fermions have zero modes corresponding to the
topology of the graph, which can be understood as kernels of the incidence
matrices of the graph and the dual graph. In the continuous theory, a scalar
curvature appears as an anomaly in the Ward-Takahashi (WT) identity associated
with a U(1) symmetry. We find that the same anomaly arises as the deficit angle
at each vertex on the graph. By using the localization method, we show that the
path integral on the graph reduces to an integral over a set of the zero modes.
The partition function is then ill-defined unless suitable operators are
inserted. We extend the same argument to the non-Abelian theory and show that
the path integral reduces to multiple integrals of Abelian theories at the
localization fixed points.
