The random-cluster model is a dependent percolation model that has
applications in the study of Ising and Potts models. In this paper, several new
results are obtained for the random-cluster model on nonamenable graphs with
cluster parameter $q\geq 1$. Among these, the main ones are the absence of
percolation for the free random-cluster measure at the critical value, and
examples of planar regular graphs with regular dual where $\pc^\f (q) > \pu^\w
(q)$ for $q$ large enough. The latter follows from considerations of
isoperimetric constants, and we give the first nontrivial explicit calculations
of such constants. Such considerations are also used to prove non-robust phase
transition for the Potts model on nonamenable regular graphs.