  In this paper, the investigation is first motivated by showing two examples
of simple regular symmetrical graphs, which have the same structural
parameters, such as average distance, degree distribution and node betweenness
centrality, but have very different synchronizabilities. This demonstrates the
complexity of the network synchronizability problem. For a given network with
identical node dynamics, it is further shown that two key factors influencing
the network synchronizability are the network inner linking matrix and the
eigenvalues of the network topological matrix. Several examples are then
provided to show that adding new edges to a network can either increase or
decrease the network synchronizability. In searching for conditions under which
the network synchronizability may be increased by adding edges, it is found
that for networks with disconnected complementary graphs, adding edges never
decreases their synchronizability. This implies that better understanding and
careful manipulation of the complementary graphs are important and useful for
enhancing the network synchronizability. Moreover, it is found that an
unbounded synchronized region is always easier to analyze than a bounded
synchronized region. Therefore, to effectively enhance the network
synchronizability, a design method is finally presented for the inner linking
matrix of rank 1 such that the resultant network has an unbounded synchronized
region, for the case where the synchronous state is an equilibrium point of the
network.
