(0,0)(50,10) (5,5)(45,5) (5,5)(10,0)[5]{}[(0,0)[.7]{}]{} (5,3)[$(k-1)'$]{} (15,3)[$k$]{} (25,3)[$k'$]{} (35,3)[$k+1$]{} (45,3)[$(k+1)'$]{} (3,5)[$\cdots$]{} (48,5)[$\cdots$]{} (10,6)[$\Pi'$]{} (20,6)[$\Pi$]{} (30,6)[$\Pi'$]{} (40,6)[$\Pi$]{}

where $\Pi=I_r$ and $\Pi' = \{r+1,...,2r\}$. The cluster variables and mutation matrices at the nodes $k$, $k'$ are given by Theorem \[simple\].

The $Q$-system for the algebras $B_r,C_r,F_4$
---------------------------------------------

In the case of non-simply laced algebras, we describe the graph ${\mathcal G}_{{\mathfrak{g}}}$ for algebras of types $B_r,C_r,F_4$ separately from $G_2$.

Let ${{\mathfrak{g}}}$ be a simple algebra of type $B_r$, $C_4$ or $F_4$. In the Dynkin diagram of ${{\mathfrak{g}}}$, let ${{\ell}}$ be the long root connected to the short root ${{s}}$:

(0,0)(50,5) (0,2)[$B_r:$]{} (11,2)(19,2) (25,2)[$\cdots$]{} (31,2)(39,2) (41,2)(49,2) (10,2)(10,0)[5]{}[(0,0)[1]{}]{} (10,0)[1]{} (20,0)[2]{} (30,0)[$r-2$]{} (40,0)[$r-1$]{} (50,0)[$r$]{} (50,2)[$s$]{} (40,2)[$\ell$]{}

(0,0)(50,5) (0,2)[$C_r:$]{} (11,2)(19,2) (25,2)[$\cdots$]{} (31,2)(39,2) (41,2)(49,2) (10,2)(10,0)[5]{}[(0,0)[1]{}]{} (10,0)[1]{} (20,0)[2]{} (30,0)[$r-2$]{} (40,0)[$r-1$]{} (50,0)[$r$]{} (50,2)[${{\ell}}$]{} (40,2)[${{s}}$]{}

(0,0)(40,5) (0,2)[$F_4:$]{} (11,2)(19,2) (21,2)(29,2) (31,2)(39,2) (10,2)(10,0)[4]{}[(0,0)[1]{}]{} (10,0)[1]{} (20,0)[2]{} (30,0)[3]{} (40,0)[4]{} (20,2)[${{\ell}}$]{} (30,2)[${{s}}$]{}

.1in In each case, in the normalized $Q$-system we have $$\begin{aligned}
\widetilde{{\mathcal T}}_{2k+1}^{({{s}},{{\ell}})} &=& {{R}}_{{{\ell}},k}{{R}}_{{{\ell}},k+1} =
{{R}}_{{{\ell}},k}^{[-C_{{{s}}{{\ell}}}/2]_+} {{R}}_{{{\ell}},k+1}^{[-C_{{{s}}{{\ell}}}/2]_+} 
;\label{odd}\\
\widetilde{{\mathcal T}}_{2k}^{({{s}},{{\ell}})} &=& {{R}}_{{{\ell}},k}^2 = {{R}}_{{{\ell}},k}^{[-C_{{{s}}{{\ell}}}]_+}
;\label{even}\\
\widetilde{{\mathcal T}}_{k}^{({{\ell}},{{s}})} &=& {{R}}_{{{s}},2k} =
{{R}}_{{{s}},2k}^{[-C_{{{\ell}},{{s}}}]_+}\label{shlo}.\end{aligned}$$ In case $({{\alpha}},\beta)$ are not the pair $({{\ell}},{{s}})$ or $({{s}},{{\ell}})$ then $$\label{otherwise}
\widetilde{{\mathcal T}}_k^{({{\alpha}},\beta)}= {{R}}_{\beta,k}^{[-C_{\beta,{{\alpha}}}]_+}.$$ Define the sets

-   $\Pi'=I_r'= \{r+1,...,2r\}$;

-   $\Pi_<$: The subset in $I_r$ corresponding to the short roots of ${{\mathfrak{g}}}$;

