# Number of linearly independent eigenvectors of a directed weighted graph Laplacian

Define the Laplacian matrix as $$L = D - A$$. Here, $$A$$ is the adjacency matrix of a directed weighted graph with $$n$$ vertices so that the entries $$A_{ij}$$ of $$A$$ are equal to a positive weight if there is an arrow form the vertex $$j$$ to $$i$$ and $$0$$ otherwise, and $$D = \operatorname{diag}(\sum_{i=1}^n A_{i1},\cdots,\sum_{i=1}^n A_{in})$$. In some applications one takes the transpose of $$L$$.

Can anything be said about the number of linearly independent eigenvectors of $$L$$? There is no guarantee that $$L$$ has $$n$$ linearly independent eigenvectors. See [Provide an example of weighted directed graph with defective Laplacian matrix for an example.

Is there any relatively simple rule which can tell under which circumstances a graph has a defective Laplacian? Here, by "defective matrix" I mean a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable.