On singularity and properties of eigenvectors of complex Laplacian matrix of multidigraphs

In this article, we associate a Hermitian matrix to a multidigraph G. We call it the complex Laplacian matrix of G and denote it by L​C(G). It is shown that the complex Laplacian matrix is a generalization of the Laplacian matrix of a graph. But, unlike the Laplacian matrix of a graph, the complex Laplacian matrix of a multidigraph may not always be singular. We obtain a necessary and sufficient condition for the complex Laplacian matrix of a multidigraph to be singular. For a multidigraph G, if L​C(G) is singular, we say G is L​C-singular. We generalize some properties of the Fiedler vectors of undirected graphs to the eigenvectors corresponding to the second smallest eigenvalue of L​C-singular multidigraphs.