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Consider a cyclic directed graph $G$, is it true that the power series computation of its adjacency matrix $M$ can stop after $k$ steps, being $k$ the directed diameter of $G$, "converging" to a point in which every node in the graph has been reached with any other?

I try to prove this considering the linear independence of the first $k$ powers of $M$. Am I wrong or this holds for directed cyclic graphs too?

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