Synchronization on star-like graphs and emerging Zp symmetries at strong coupling

We discuss the aspects of synchronization on inhomogeneous star-like graphs with long rays in the Kuramoto model framework. We assume the positive correlation between internal frequencies and degrees for all nodes which supports the abrupt first-order synchronization phase transition. It is found that different ingredients of the graph get synchronized at different critical couplings. Combining numerical and analytic tools, we evaluate all critical couplings for the long star graph. Surprisingly, it is found that at strong coupling there are discrete values of coupling constant that support the synchronized states with emerging Z(p) symmetries. The stability of the synchronized phase is discussed, and the interpretation of the phase with emerging Z(p) symmetry for the Josephson array on a long star graph is mentioned.