  Synchronization in networks of coupled oscillators is known to be largely
determined by the spectral and symmetry properties of the interaction network.
Here we leverage this relation to study a class of networks for which the
threshold coupling strength for global synchronization is the lowest among all
networks with the same number of nodes and links. These networks, defined as
being uniform, complete, and multi-partite (UCM), appear at each of an infinite
sequence of network-complement transitions in a larger class of networks
characterized by having near-optimal thresholds for global synchronization. We
show that the distinct symmetry structure of the UCM networks, which by design
are optimized for global synchronizability, often leads to formation of
clusters of synchronous oscillators, and that such states can coexist with the
state of global synchronization.
