Let $G$ be a digraph with adjacency matrix $A =(A_{ij})$ where $A_{ij}=1$ if and only if there is a directed edge $i \to j$ and $A_{ij}=0$ otherwise. Let $D= (D_{ij})$ be the degree matrix with $D_{ij} = 0$ if $i \neq j$ and $D_{ii} = \sum_j A_{ij}$. Define the digraph Laplacian $L=D-A$. For which digraphs are all eigenvalues of $L$ real with exactly one negative?
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$\begingroup$ Eigenvalues of digraphs might well be complex; e.g. there are examples where they are either complex or positive. $\endgroup$– Dima PasechnikCommented Jan 31, 2016 at 21:01
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$\begingroup$ Even if you assume $G$ is regular, it is not easy to see how to decide if its eigenvalues are real. $\endgroup$– Chris GodsilCommented Feb 1, 2016 at 22:36
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1 Answer
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By the Gershgorin circle theorem, all eigenvalue of the Laplacian matrix have non-negative real parts.
