  The Kuramoto model is a system of nonlinear differential equations that
models networks of coupled oscillators and is often used to study
synchronization among them. It has been observed that if the natural
frequencies of the oscillators are similar they will phase-lock, meaning that
they oscillate at a common frequency with fixed phase differences. Conversely,
we do not observe this behavior when the natural frequencies are very
dissimilar. Previously, Bronski and the author gave upper and lower bounds for
the volume of the set of frequencies exhibiting phase-locking behavior. This
was done under the assumption that any two oscillators affect each other with
equal strength. In this paper the author generalizes these upper and lower
bounds by removing this assumption. Similar to the previous work with Bronski,
where the upper and lower bounds are sums over spanning trees of the network,
our generalized upper and lower bounds are sums over certain directed subgraphs
of the network. In particular, our lower bound is a sum over directed spanning
trees. Finally, we numerically simulate the dependence of the number of
directed spanning trees on the presence of certain two edge motifs in the
network and compare this dependence with that of the synchronization of
oscillator models found by Nykamp, Zhao, and collaborators.
