call this specialization of the initial parameters the KR point.

On the other hand, it is known that cluster variables obey the Laurent phenomenon [@FZlaurent]. We show that, under the specialization of the cluster variables to the KR point, this becomes what we called in [@Ke07] the strong Laurent phenomenon. That is, the cluster variables on the entire cluster graph, not just the subgraph corresponding to the $Q$-system, are polynomials in the initial cluster variables.

The paper is organized as follows. We recall the definition of $Q$-systems for any simple Lie algebra in Section 2, as well as the definition of normalized cluster algebras without coefficients. Section 3 deals with the formulation of each $Q$-system as a subgraph in a cluster algebra. In Section 3.1, we review the results of [@Ke07] about the formulation of $Q$-systems as cluster algebras for simply-laced Lie algebras. In Sections 3.2 and 3.3, we formulate the cluster algebras corresponding to the non-simply laced simple Lie algebras. In Section 4, we prove that the special boundary conditions which give solutions of the $Q$-system as characters of Kirillov-Reshetikhin modules imply the polynomiality of the cluster variables as functions of the seed variables at the boundary node. Section 