  We develop the mutation theory in the exact WKB analysis using the framework
of cluster algebras. Under a continuous deformation of the potential of the
Schr\"odinger equation on a compact Riemann surface, the Stokes graph may
change the topology. We call this phenomenon the mutation of Stokes graphs.
Along the mutation of Stokes graphs, the Voros symbols, which are monodromy
data of the equation, also mutate due to the Stokes phenomenon. We show that
the Voros symbols mutate as variables of a cluster algebra with surface
realization. As an application, we obtain the identities of Stokes
automorphisms associated with periods of cluster algebras. The paper also
includes an extensive introduction of the exact WKB analysis and the surface
realization of cluster algebras for nonexperts.
