$n+1$ equilibrium equations must sum to zero. This allows the elimination of one of the equations, producing the system  of $n$ equations in $n$ unknowns. Despite its mechanical origin, the frequency synchronization system  naturally appears in a long list of seemingly unrelated fields, including electrical power networks [@Baillieul1982; @dorfler_2013], flocking behavior in biology and control theory [@Justh2004; @Vicsek1995], and decentralized clock synchronization [@Simeone2008]. We refer to [@dorfler_synchronization_2014] for a detailed list.

In [@Baillieul1982], an upper bound on the number of equilibria of the Kuramoto model (solutions to ) induced by a graph of $N$ vertices with any coupling strengths is shown to be $\binom{2N-2}{N-1}$. For certain cases such as the Kuramoto model on the one, two and three-dimensional lattice graphs with different boundary conditions, as well as for complete and planar graphs, all or at least a class of equilibria were analytically [@Casetti:June2003:0022-4715:1091; @delabays2016multistability; @delabays2017multistability; @kastner2011stationary; @mehta2011stationary; @Nerattini:2012pi; @ochab2010synchronization; @xin2016analytical] and numerically [@Hughes:2012hg; @hughes2014inversion; @manik2016cycle; @Mehta:2013iea; @xi2017synchronization] found in previous studies. For tree graphs of $N$ nodes, it is well known that there could be as many as $2^{N-1}$ real equilibria. Various algebraic formulations have been used to leverage results from algebraic geometry and numerically find some or all equilibria 