  We study the emergence of synchronization in the Kuramoto model on a digraph
in the presence of time delays. Assuming the digraph is strongly connected, we
first establish a uniform bound on the phase diameter and subsequently prove
the asymptotic frequency synchronization of the oscillators under suitable
assumptions on the initial configurations. In the case of an all-to-all
connection, we obtain an exponential synchronization estimate. Additionally, we
present numerical simulations, providing further insights into the
synchronization and oscillatory behaviors of the oscillator frequencies
depending on the network structure and the magnitude of the time delay.
