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I consider (weighted) directed graph and eigenvalues of its laplacian matrix. If a graph contains rooted out-branching which is the subgraph possessing a node can approaching to any nodes in the graph, then laplacian matrix has only one 0 eigenvalue.

The question is "how many multiplicity of 0 eigenvalue of directed laplacian matrix for weakly connected graph does exists" or "how can we identify it".

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The rank of Laplacian is $n-\omega(G)$ where $\omega(G)$ is the number of connected components. That is, $\lambda_{n-1}\neq0$ iff the graph is connected. See p. 147

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