x}}[t],B[t])$ where we have the following relation (mutations or evolution equations along the edge $k$) between the cluster at $t$ and $t'$: $$\begin{aligned}
\mu_k(x_i) &=& \left\{ \begin{array}{ll} x_i, & i\neq k; \\
x_i^{-1}(\prod_j x_j^{[B_{jk}]_+} + \prod_{j} x_j^{[-B_{jk}]_+}). &
i=k.\end{array}\right.\\
\mu_k(B_{ij}) &=& \left\{ \begin{array}{ll}
-B_{ij}, & \hbox{if $i=k$ or $j=k$}; \\
B_{ij} + {\rm sgn}(B_{ik})[B_{ik}B_{kj}]_+, &
\hbox{otherwise}.\end{array}\label{Bmutations}\right. \end{aligned}$$ where $[n]_+$ denotes the positive part of $n$ and $${{\rm sgn}}(n)=\left\{ \begin{array}{ll} 0, & n=0;\\
1, & n>0; \\
-1, & n<0.\end{array}\right.$$

One can check the following properties of cluster mutations:

-   The mutations preserve the skew-symmetry of $B$.

-   If $B_{ij}=0$ then the mutations $\mu_i$ and $\mu_j$ commute.

-   If $B_{ij}=0$ and $k,l$ are two integers distinct from both $i$ and $j$, then $$\label{commutingcompound}
    \mu_i\circ \mu_j( B_{kl}) = B_{kl} + {{\rm sgn}}(B_{ki})[B_{ki}B_{il}]_+
    + {{\rm sgn}}(B_{kj})[B_{kj}B_{jl}]_+.$$ This statement can be extended to any finite sequence of commuting mutations.

-   Mutations are defined in such a way that $\mu_i^2=1$. Thus, cluster graphs are not oriented.

We define ${\mathcal A}$ to be the family of variables $\{{{R}}_{{{\alpha}},i}\}_{i\in {{\mathbb Z}},{{\alpha}}\in I_r}$ which satisfy the normalized $Q$-system .

In what follows we are interested in subgraphs of $\mathbb T_n$ which encode the $Q$-system, that is, the cluster 