  The zero temperature quenching dynamics of the ferromagnetic Ising model on a
densely connected small world network is studied where long range bonds are
added randomly with a finite probability $p$. We find that in contrast to the
sparsely connected networks and random graph, there is no freezing and an
initial random configuration of the spins reaches the equilibrium configuration
within a very few Monte Carlo time steps in the thermodynamic limit for any $p
\ne 0$. The residual energy and the number of spins flipped at any time shows
an exponential relaxation to equilibrium. The persistence probability is also
studied and it shows a saturation within a few time steps, the saturation value
being 0.5 in the thermodynamic limit. These results are explained in the light
of the topological properties of the network which is highly clustered and has
a novel small world behaviour.
