In this paper we extend our previous results on the connectivity functions
and pressure of the Random Cluster Model in the highly subcritical phase and in
the highly supercritical phase, originally proved only on the cubic lattice
$\Z^d$, to a much wider class of infinite graphs. In particular, concerning the
subcritical regime, we show that the connectivity functions are analytic and
decay exponentially in any bounded degree graph. In the supercritical phase, we
are able to prove the analyticity of finite connectivity functions in a smaller
class of graphs, namely, bounded degree graphs with the so called minimal
cut-set property and satisfying a (very mild) isoperimetric inequality. On the
other hand we show that the large distances decay of finite connectivity in the
supercritical regime can be polynomially slow depending on the topological
structure of the graph. Analogous analyticity results are obtained for the
pressure of the Random Cluster Model on an infinite graph, but with the further
assumptions of amenability and quasi-transitivity of the graph.