Laplacian matrices of general complex weighted directed graphs

We introduce the concept of general complex weighted directed graphs where each edge is assigned a complex number. Necessary and sufficient conditions for the Laplacian matrix to be singular/nonsingular are derived. Our results give the relationship between the Laplacian matrix and the structure of its corresponding directed graph. Compared with the existing results, our main contribution is that our results are established without the restriction that the adjacency matrix is Hermitian.