  A cluster algebra is an algebraic structure generated by operations of a
quiver (a directed graph) called the mutations and their associated simple
birational mappings. By using a graph-combinatorial approach, we present a
systematic way to derive a tropical, i.e. subtraction-free birational,
representation of Weyl groups from cluster algebras. Our result provides a
broad class of Weyl group actions including previously known examples acting on
certain rational varieties [28, 30] and hence it is relevant to q-Painleve
equations and their higher-order extensions. Key ingredients of the argument
are the combinatorial aspects of the reflection associated with a cycle
subgraph in the quiver. We also discuss symplectic structures of the discrete
dynamical systems thus obtained.
