laplacian_matrix

laplacian_matrix(G, nodelist=None, weight='weight')

Returns the Laplacian matrix of G.

The graph Laplacian is the matrix L = D - A, where A is the adjacency matrix and D is the diagonal matrix of node degrees.

Parameters:

Ggraph

A NetworkX graph

nodelistlist, optional

The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes().

weightstring or None, optional (default=’weight’)

The edge data key used to compute each value in the matrix. If None, then each edge has weight 1.

Returns:

LSciPy sparse array

The Laplacian matrix of G.

See also

to_numpy_array()

normalized_laplacian_matrix

laplacian_spectrum()

Notes

For MultiGraph, the edges weights are summed.

Examples

For graphs with multiple connected components, L is permutation-similar to a block diagonal matrix where each block is the respective Laplacian matrix for each component.

>>> G = nx.Graph([(1, 2), (2, 3), (4, 5)])
>>> print(nx.laplacian_matrix(G).toarray())
[[ 1 -1  0  0  0]
 [-1  2 -1  0  0]
 [ 0 -1  1  0  0]
 [ 0  0  0  1 -1]
 [ 0  0  0 -1  1]]