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For an undirected graph $G$, its adjacency matrix $A_G$ is symmetric, and by the (consequence of) spectral theorem, each of its Jordan blocks has size $1$. This is not true for a general directed graph, except for certain special cases like the directed cycle graphs.

Question: Suppose in a directed graph $G$, each edge is contained in a cycle (e.g., undirected graphs, directed cycle graphs). Is it true that each Jordan block of $A_G$ has size $1$?

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It is not true. Take the digraph with adjacency matrix $$A=\begin{bmatrix} 0& 1 &0& 1\\ 0 & 0 & 1& 0\\ 0 & 1& 0 &1\\ 1 & 0 &0&0\end{bmatrix}.$$ This digraph is strongly connected (so every edge is contained in a cycle). The eigenvalues are $0,0,\pm\sqrt{2}$ and the Jordan block of $0$ is $2\times 2$.

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  • $\begingroup$ Thank you for the example. $\endgroup$
    – F J
    Commented Aug 18, 2023 at 2:40

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