role network topology plays in the exhaustive study of synchronization in Kuramoto model. From a computational view point, these generically exact explicit upper bounds on the number of solutions are also of great importance in numerical methods for finding all synchronization configurations for the Kuramoto model: It provides an explicit stopping criteria for iterative solvers such as Newton-based solvers as well as the homotopy-based Monodromy method [@Duff2016]. The secondary contribution is the general approach of computing the birationally invariant intersection index by finding the appropriate relaxation: Using the much simpler construction of the adjacency polytope bound, the problem is transformed into a problem of computing normalized volumes for certain polytopes. The rest of the paper is structured as follows. In §\[sec:background\], we briefly review the Kuramoto model and existing results on the number of possible equilibria. §\[sec:prelim\] reviews notations and well known theorems to be used. In §\[sec:tree\] and §\[sec:cycle\], we compute $[L_{G,1},\dots,L_{G,n}]$ for trees and cycles respectively.

Kuramoto model and synchronization equations {#sec:background}
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(0,0) circle(5.0); (5,0) – (0,5); (0,5) – (-3,-4); (5,0) – (3,-4); (3,-4) – (-3,-4); ( 5, 0) arc\[radius=5, start angle=0 , end angle=20\]; ( 0, 5) arc\[radius=5, start angle=90 , end angle=110\]; (-3,-4) arc\[radius=5, 