  We prove a sufficient condition for synchronization for coupled
one-dimensional maps and estimate the size of the window of parameters where
synchronization takes place. It is shown that coupled systems on graphs with
positive eigenvalues (EVs) of the normalized graph Laplacian concentrated
around 1 are more amenable for synchronization. In the light of this condition,
we review spectral properties of Cayley, quasirandom, power-law graphs, and
expanders and relate them to synchronization of the corresponding networks. The
analysis of synchronization on these graphs is illustrated with numerical
experiments. The results of this paper highlight the advantages of random
connectivity for synchronization of coupled chaotic dynamical systems.
