be expressed as the generalized mixed volume of the Newton-Okounkov bodies associated with $L_{G,1},\dots,L_{G,n}$, its direct computation, in general, remains a difficult problem. Using a construction known as the “adjacency polytope bound” developed in [@chen_unmixing_2017; @chen_network_2015], the primary contribution of this paper is the computation of explicit formulas for the birationally invariant intersection index $[L_{G,1},\dots,L_{G,n}]$ for certain graphs. In particular, we show that for trees and cycles of $N$ vertices, the intersection index $[L_{G,1},\dots,L_{G,n}]$ is $2^{N-1}$ and $N \binom{N-1}{\lfloor (N-1)/2 \rfloor}$ respectively. Both are significantly less than the only known upper bound $\binom{2N-2}{N-1}$ for the general case (hetergeneous oscillators with nonuniform coupling) of the Kuramoto equations  even for small values of $n$. Asymptotically, in both cases, the ratio between the new bounds and $\binom{2N-2}{N-1}$ goes to zero as $N \to \infty$. Moreover, the intersection index derived from tree graphs also coincide with the well known lower bound of the number of *real* solutions to the original (non-algebraic) system  showing that the intersection index derived from a complex solution bound can actually be attained by real solutions. These results are dramatic improvements over the existing bound on the number of synchronization configuration for a Kuramoto model. They also confirm the crucial 