The two- and three-dimensional q-state Potts model is simulated by a cluster algorithm and the relaxation dynamics of average cluster size and the order parameter during initial thermalization are studied. The dependence of these quantities on the lattice size and Ci in the early stages of Monte Carlo iterations reveals valuable information on the nature of the phase transition occurring in the system. From the initial thermalization data of the 2D 7-state Potts model, surface tension is also calculated. The value is in good agreement with the exact result. (C) 1998 Elsevier Science B.V. All rights reserved.