Let $\Delta$ be an oriented valued graph equipped with a group of admissibleautomorphisms satisfying a certain stability condition. We prove that the(coefficient-free) cluster algebra $\mathcal A(\Delta/G)$ associated to thevalued quotient graph $\Delta/G$ is a subalgebra of the quotient $\pi(\mathcalA(\Delta))$ of the cluster algebra associated to $\Delta$ by the action of $G$.When $\Delta$ is a Dynkin diagram, we prove that $\mathcal A(\Delta/G)$ and$\pi(\mathcal A(\Delta))$ coincide. As an example of application, we prove thataffine valued graphs are mutation-finite, giving an alternative proof to aresult of Seven.
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