the analysis of the Kuramoto model on cycle graphs is also of great practical importance [@xi2017synchronization].

For a cycle graph $C_N$ of $N = n + 1$ vertices (labeled by $\{0,\dots,n\}$), we shall show the intersection index $[L_{C_N,1},\dots,L_{C_N,n}]$ is $(n+1)\binom{n}{\lfloor n/2 \rfloor}$. Following the same strategy used in the previous section, we first compute the AP bound for the cycle graph $C_N$. Then we show there is no gap between $[L_{C_N,1},\dots,L_{C_N,n}]$ and the AP bound.

The set of edges is $E(C_N) = \{(0,1),(1,2),\dots,(n-1, n),(n,0)\}$. The induced adjacency polytope (Definition \[def:adj-polytope\]) is $$\nabla_{C_N} = \operatorname{conv}\{\pm(\mathbf{e}_i - \mathbf{e}_j) \mid (i,j) \in E(C_N)\},$$ where $\mathbf{e}_0 = (0,\dots,0)$ as before. The AP bound for $F_{C_N}$ is defined to be the normalized volume of $\nabla_{C_N}$; thus, the first goal of this section will be to identify this normalized volume. It will be simplest to first notice that $\nabla_{C_N}$ is unimodularly equivalent to the polytope $$P_N = \operatorname{conv}\{
        \pm \mathbf{e}_1 \, , \,\dots \, , \, \pm \mathbf{e}_n\,,\;
        \pm (\mathbf{e}_1 + \cdots + \mathbf{e}_n)
        \}.$$ Such an equivalence can be seen by applying the normalized volume-preserving transformation given by $$\begin{bmatrix}
        1 & 0 & 0 & \cdots & 0 \\
        1 & 1 & 0 & 