# scipy.sparse.csgraph.laplacian¶

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

Return the Laplacian matrix of a directed graph.

Parameters
csgrapharray_like or sparse matrix, 2 dimensions

compressed-sparse graph, with shape (N, N).

normedbool, optional

If True, then compute normalized Laplacian.

return_diagbool, optional

If True, then also return an array related to vertex degrees.

use_out_degreebool, optional

If True, then use out-degree instead of in-degree. This distinction matters only if the graph is asymmetric. Default: False.

Returns
lapndarray or sparse matrix

The N x N laplacian matrix of csgraph. It will be a numpy array (dense) if the input was dense, or a sparse matrix otherwise.

diagndarray, optional

The length-N diagonal of the Laplacian matrix. For the normalized Laplacian, this is the array of square roots of vertex degrees or 1 if the degree is zero.

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.

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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