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What are the necessary and sufficient conditions for a PSD matrix $S$, to be a graph Laplacian? I know $S1=0$ is required. But clearly a real zero sum PSD matrix is not necessarily a graph Laplacian.

Second question arises after seeing the useful responses here: What are the conditions for a PSD matrix to be a weighted graph Laplacian?

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    What is a PSD matrix?2012-10-05
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    Sorry, I meant positive semi-definite matrix.2012-10-05

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