# scipy.sparse.csgraph.laplacian¶

scipy.sparse.csgraph.laplacian(csgraph, normed=False, return_diag=False)[source]

Return the Laplacian matrix of a directed graph.

For non-symmetric graphs the out-degree is used in the computation.

Parameters : csgraph : array_like or sparse matrix, 2 dimensions compressed-sparse graph, with shape (N, N). normed : bool, optional If True, then compute normalized Laplacian. return_diag : bool, optional If True, then return diagonal as well as laplacian. lap : ndarray The N x N laplacian matrix of graph. diag : ndarray The length-N diagonal of the laplacian matrix. diag is returned only if return_diag is True.

Notes

The Laplacian matrix of a graph is sometimes referred to as the “Kirchoff matrix” or the “admittance matrix”, and is useful in many parts of spectral graph theory. In particular, the eigen-decomposition of the laplacian matrix can give insight into many properties of the graph.

For non-symmetric directed graphs, the laplacian is computed using the out-degree of each node.

Examples

```>>> from scipy.sparse import csgraph
>>> G = np.arange(5) * np.arange(5)[:, np.newaxis]
>>> G
array([[ 0,  0,  0,  0,  0],
[ 0,  1,  2,  3,  4],
[ 0,  2,  4,  6,  8],
[ 0,  3,  6,  9, 12],
[ 0,  4,  8, 12, 16]])
>>> csgraph.laplacian(G, normed=False)
array([[  0,   0,   0,   0,   0],
[  0,   9,  -2,  -3,  -4],
[  0,  -2,  16,  -6,  -8],
[  0,  -3,  -6,  21, -12],
[  0,  -4,  -8, -12,  24]])
```

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