  Previous research claimed or disclaimed the role of a small diameter in the
synchronization of a network of coupled dynamical systems. We investigate this
connection and show that it is two folds. We first construct two classes of
networks, the polarized networks and the random networks with a fixed diameter,
which exhibit very different synchronizability. This shows that the diameter
itself is insufficient to determine the synchronizability of networks.
Secondly, we derive analytic estimates on the synchronizability of networks in
terms of the diameter, and find that a larger size of network admits of a more
flexible synchronizability. The analysis is confirmed by numerical results.
