theorem:

There exists a cluster graph ${\mathcal G}_{{\mathfrak{g}}}$, which includes all nodes labeled by $k\in {{\mathbb Z}}_+$, with corresponding cluster variables ${{\mathbf x}}[k]$ as in and $B$ as in , such that all mutations in the graph are $Q$-system evolutions. Moreover, the set ${\mathcal A}$ is equal to the union of the cluster variables over all nodes in ${\mathcal G}_{{\mathfrak{g}}}$.

We will describe such a graph for each algebra ${{\mathfrak{g}}}$ below.

In all cases considered below, if the mutations $\{\mu_{i_1},...,\mu_{i_n}\}$ commute starting from node $t$ (that is, if $B_{i_k,i_j}=0$ for all $k,j$ at the node $t$), then we may act with these mutations in any order, as long as they are distinct, and nodes corresponding to acting with a certain set of mutations, regardless of order, can be identified with each other in the quotient graph ${\mathcal G}_{{\mathfrak{g}}}$ (this is similar in spirit to the bipartite graph of [@FZfour]). These mutations give rise to an n-simplex (e.g. a cube in the case $n=3$) in ${\mathcal G}_{{\mathfrak{g}}}$.

We will in general describe only the cluster variables reached via certain compound mutations $\mu^{(i)}=\mu_{i_1}\circ\cdots\circ\mu_{i_n}$, where the corresponding mutations commute. The cluster graph ${\mathcal G}_{{\mathfrak{g}}}$ is recovered by replacing a compound mutation with the 