I am concerned with unweighted directed graphs where each node contains exactly one edge pointing to another node, which could be itself. In other words, each row of the adjacency matrix contains one entry equal to 1, and the rest are 0. Im not sure if these graphs have a name, but they could be called a deterministic Markov chain.

I think the eigenvalues of the adjacency matrix will always come from this set: 1,0 and roots of unity. Is this true, and if so, what ensures this?