Classification of sectors of the Cuntz algebras by graph invariants

A unitary equivalence class of endomorphisms of a unital C${}^{\ast }$-algebra $\mathcal{A}$ is called a {\it sector} of $\mathcal{A}$. We introduced permutative endomorphisms of the Cuntz algebra ${\mathcal{O}}_N$ in the previous work. Branching laws of permutative representations of ${\mathcal{O}}_N$ by them are computed by directed regular graphs. In this article, we classify sectors associated with permutative endomorphisms of ${\mathcal{O}}_N$ by their graph invariants concretely.