show that any directed cycle graph $C_n$ will be uniquely determined by its spectrum of adjacency matrix for directed graph.
it is easy to see that the eigenvalues for directed cycle is $e^{\frac{2\pi ri}{n}} $for $0\leq r\leq n-1$ .
now suppose we have directed graph which its spectrum is what we define above,and we suppose that it is not directed cycle,now I have problem to make contradiction, it will be great if you give me hint,Idea or reference to study ,thanks.