  In this article we study synchronization of Kuramoto oscillators with
heterogeneous frequencies, and where underlying topology is a graph of diameter
two. When the coupling strengths between every two connected oscillators are
the same, we find an analytic condition that guarantees an existence of a
Positively Invariant Set (PIS) and demonstrate that existence of a PIS suffices
for frequency synchronization. For graphs of diameter two, this synchronization
condition is significantly better than existing general conditions for an
arbitrary topology. If the coupling strengths can be different for different
pairs of connected oscillators, we formulate an optimization problem that finds
sufficient for synchronization coupling strengths such that their sum is
minimal.
