On Exchange Spectra of Valued Cluster Quivers and Cluster Algebras

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.