  Synchronization in coupled oscillators networks is a remarkable phenomenon of
relevance in numerous fields. For Kuramoto oscillators the loss of
synchronization is determined by a trade-off between coupling strength and
oscillator heterogeneity. Despite extensive prior work, the existing sufficient
conditions for synchronization are either very conservative or heuristic and
approximate. Using a novel cutset projection operator, we propose a new family
of sufficient synchronization conditions; these conditions rigorously identify
the correct functional form of the trade-off between coupling strength and
oscillator heterogeneity. To overcome the need to solve a nonconvex
optimization problem, we then provide two explicit bounding methods, thereby
obtaining (i) the best-known sufficient condition for unweighted graphs based
on the 2-norm, and (ii) the first-known generally-applicable sufficient
condition based on the $\infty$-norm. We conclude with a comparative study of
our novel $\infty$-norm condition for specific topologies and IEEE test cases;
for most IEEE test cases our new sufficient condition is one to two orders of
magnitude more accurate than previous rigorous tests.
