A generalization of de Bruijn graphs and classification of endomorphisms of Cuntz algebras by graph invariants

The classification problem of endomorphisms of the Cuntz algebra O_Nmathcal{O}_{N} is solved by using graph theory. We introduce permutative de Bruijn graphs as generalizations of de Bruijn graphs. Branching laws for a permutative endomorphism ρ of O_Nmathcal{O}_{N} are computed by using the permutative de Bruijn graph associated with ρ. According to this correspondence between endomorphisms and graphs, we classify permutative endomorphisms of O_Nmathcal{O}_{N} by graph invariants concretely.