  Inspirited by the importance of the spectral theory of graphs, we introduce
the spectral theory of valued cluster quiver of a cluster algebra. Our aim is
to characterize a cluster algebra via its spectrum so as to use the spectral
theory as a tool.
  First, we give the relations between exchange spectrum of a valued cluster
quiver and adjacency spectrum of its underlying valued graph, and between
exchange spectra of a valued cluster quiver and its full valued subquivers. The
key point is to find some invariants from the spectrum theory under mutations
of cluster algebras, which is the second part we discuss. We give a sufficient
and necessary condition for a cluster quiver and its mutation to be cospectral.
Following this discussion, the so-called cospectral subalgebra of a cluster
algebra is introduced. We study bounds of exchange spectrum radii of cluster
quivers and give a characterization of $2$-maximal cluster quivers via the
classification of oriented graphs of its mutation equivalence. Finally, as an
application of this result, we obtain that the preprojective algebra of a
cluster quiver of Dynkin type is representation-finite if and only if the
cluster quiver is $2$-maximal.
