  For the high-dimensional Kuramoto model with identical oscillators under a
general digraph that has a directed spanning tree, although exponential
synchronization was proved under some initial state constraints, the exact
exponential synchronization rate has not been revealed until now. In this
paper, the exponential synchronization rate is precisely determined as the
smallest non-zero real part of Laplacian eigenvalues of the digraph. Our
obtained result extends the existing results from the special case of strongly
connected balanced digraphs to the condition of general digraphs owning
directed spanning trees, which is the weakest condition for synchronization
from the aspect of network structure. Moreover, our adopted method is
completely different from and much more elementary than the previous
differential geometry method.
