  We examine the onset of synchronization transition in a star network of
Kuramoto phase oscillators in the presence of inertia and a time delay in the
coupling. A direct correlation between the natural frequencies of the
oscillators and their degrees is assumed. The presence of time delay is seen to
enhance the onset of first order synchronization. The star network also
exhibits different synchronization transitions depending on the value of time
delay. An analytical prediction to observe the effect of the time delay is
provided and further supported by simulation results. Our findings may help
provide valuable insights into the understanding of mechanisms that lead to
synchronization on complex networks.
