  We study KMS states on finite-graph C*-algebras with sinks and sources. We
compare finite-graph C*-algebras with C*-algebras associated with complex
dynamical systems of rational functions. We show that if the inverse
temperature $\beta$ is large, then the set of extreme $\beta$-KMS states is
parametrized by the set of sinks of the graph. This means that the sinks of a
graph correspond to the branched points of a rational funcition from the point
of KMS states. Since we consider graphs with sinks and sources, left actions of
the associated bimodules are not injective. Then the associated graph
C*-algebras are realized as (relative) Cuntz-Pimsner algebras studied by
Katsura. We need to generalize Laca-Neshevyev's theorem of the construction of
KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules
are not injective.
