\cup \cdots \cup \nabla_{G,n}) =
        \operatorname{conv}( \{ \pm (\mathbf{e}_i-\mathbf{e}_j) \mid (i,j) \in E(G) \}).$$ The normalized volume $\operatorname{NVol}(\nabla_G)$ is called the **adjacency polytope bound** of $G$.

The polytope $\nabla_G$ can be considered as a geometric encoding of the topology of the graph $G$. Adjacency polytopes have been previously studied in order to identify properties of a related semigroup algebra, such as in [@centrallysymmetric]. However, previous work has not addressed the normalized volume of these polytopes. A simple observation [@chen_unmixing_2017; @chen_network_2015] is that the adjacency polytope bound (or simply, **AP bound**) is indeed an upper bound for answers of Problem \[prb:Cstar\] and \[prb:index\]:

\[pro:ap-bound\] Given a graph $G$ containing vertices $\{0,1,\dots,n\}$, the number of isolated ${\mathbb{C}}^*$-solutions for the algebraic system  is bounded by the AP bound $\operatorname{NVol}(\nabla_G)$.

By comparing the constructions of the solution bounds outlined above, it is easy to verify the following chain of inequalities $$\parbox{12ex}{\centering ${\mathbb{R}}$-solution\\ count of~\eqref{equ:sync-sin}}
    \;\le \;
    \parbox{12ex}{\centering ${\mathbb{C}}^*$-solution\\ count of~\eqref{equ:sync-laurent}}
    \; \le \;
    [L_{G,1},\dots,L_{G,n}]
    \; \le \;
    \parbox{7ex}{\centering BKK\\ bound} \; \le \;
    \parbox{7ex}{\centering AP\\ bound}         \label{equ:comparison}$$

Tree graphs
===========

 \[sec:tree\] This section provides the answers for Problem \[prb:Cstar\] and \[prb:index\] for a tree graph $T_N$ containing $N=n+1$ vertices. The strategy is to bound 