(4,16)[3]{} (24,16)[1]{} (44,16)[3]{} (64,16)[1]{} (16,16)[4]{} (36,16)[2]{} (56,16)[4]{} (76,16)[2]{} (14,6)[3]{} (34,6)[1]{} (54,6)[3]{} (74,6)[1]{}

The cluster algebra has four variables, and four possible mutations, which are the labels along the edges. The special nodes $k$ and their primed versions appear on the center line of the graph. The other nodes have intermediate cluster variables.

The full cluster algebra graph of ${{\mathfrak{sl}}}_3$ would include, for example, an edge labeled $4$ emanating from the node $\mu_1(k)$. However the cluster mutation in this direction is not a $Q$-system evolution.

\[Athree\] The graph ${\mathcal G}_{{{\mathfrak{sl}}}_4}$ described in Theorem \[simple\] is as follows:

(0,10)(60,33) (10,10)(0,20)(10,30) (10,10)(20,20)(10,30) (20,10)(10,20)(20,30) (20,10)(10,20)(20,30) (10,10)(20,10) (20,30)(10,30) (0,20)(10,20) (20,10)(30,20)(20,30) (40,30)(30,20)(40,10) (40,10)(50,20)(40,30) (50,10)(40,20)(50,30) (50,10)(40,20)(50,30) (20,20)(40,20) (50,10)(60,20)(50,30) (40,10)(50,10) (50,20)(60,20) (40,30)(50,30) (0,20)(10,0)[7]{}[(0,0)[.7]{}]{} (10,10)(10,0)[2]{}[(0,0)[.7]{}]{} (40,10)(10,0)[2]{}[(0,0)[.7]{}]{} (10,30)(10,0)[2]{}[(0,0)[.7]{}]{} (40,30)(10,0)[2]{}[(0,0)[.7]{}]{} (30,0)[(0,0)[.5]{}]{} (6,16)[3]{} (5,21)[2]{} (4,26)[1]{} (15,11)[2]{} (15,31)[2]{} (13,14)[1]{} (13,25)[1]{} (14,18)[3]{} (14,28)[3]{} (26,26)[3]{} (25,21)[2]{} (24,16)[1]{} (36,16)[6]{} (35,21)[5]{} (34,26)[4]{} (45,11)[5]{} (45,31)[5]{} (43,14)[4]{} (43,25)[4]{} (44,18)[6]{} (44,28)[3]{} (56,26)[6]{} (55,21)[5]{} (54,16)[4]{} (0,18)[$k$]{} (30,18)[$k'$]{} (61,18)[$k+1$]{} (63,20)[$\cdots$]{} (-3,20)[$\cdots$]{}

.1in

The graphs ${\mathcal G}_{{\mathfrak{g}}}$ which we describe are not, in general, the complete subgraph of $\mathbb T_n$ corresponding to $Q$-system evolutions, except when ${\rm rank}({{\mathfrak{g}}})\leq 2$. We chose the graphs ${\mathcal G}_{{\mathfrak{g}}}$ so that the union of all cluster variables at their nodes corresponds to the complete set of normalized 