# which kind of simple digraphs can have eigenvalue 1?

For a digraph $D$, the eigenvalues of $D$ mean the eigenvalues of its adjacency matrix. If $D$ is simple(entries of the adjacency matrix are 0 or 1), then which $D$ can have eigenvalue $r$ on the unit circle, i.e. $|r|=1$? In special, which $D$ can have 1 as an eigenvalue? As we can easily see, an acyclic digraph can not have eigenvalue 1, while a directed cycle has.

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