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Let $A$ be an $n\times n$ matrix with $a_{ii}=0$ for all $i$, and $a_{ij}\in\{0,1\}$ for all $i\neq j$, and $a_{ij}=0\leftrightarrow a_{ji}=1$ for all $i\neq j$. Is it necessary that $A$ as a unique, non-repeated eigenvalue of maximal absolute value?

I thought about applying the Perron-Frobenius Theorem, but the matrix $A$ is not necessarily irreducible, and also the Perron–Frobenius theorem for irreducible matrices doesn't seem to say that the eigenvalue of maximal absolute value is unique.

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No, this is not true. Take $\pmatrix{0 & 1 \\ 0 & 0}$, which has two zero eigenvalues; or $\pmatrix{0&1&0\\0&0&1\\1&0&0}$, which is orthogonal, hence all eigenvalues have absolute value 1.

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