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?
Assuming you mean positive-semidefinite for PSD, a graph laplacian has degrees of the graph on the diagonal, so we must have the Erdos Gallai condition for the degrees (that's necessary and sufficient for existance of the graph with given degree sequence). Other than that, the other entries must be either -1 or 0 with row and column sumbs equal to 0.