  Synchronization of identical harmonic oscillators interconnected via
position, velocity, and acceleration couplings is studied. How to construct a
complex Laplacian matrix representing the overall coupling is presented. It is
shown that the oscillators asymptotically synchronize if and only if this
matrix has a single eigenvalue on the imaginary axis. This result generalizes
some of the known spectral tests for synchronization. Some simpler Laplacian
constructions are also proved to work provided that certain structural
conditions are satisfied by the coupling graphs.
