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.
Returns: 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]])