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.


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