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normalized_laplacian_matrix(G, nodelist=None, weight='weight')[source]

Return the normalized Laplacian matrix of G.

The normalized graph Laplacian is the matrix

N = D^{-1/2} L D^{-1/2}

where L is the graph Laplacian and D is the diagonal matrix of node degrees.

  • G (graph) – A NetworkX graph
  • nodelist (list, 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().
  • weight (string 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.

N – The normalized Laplacian matrix of G.

Return type:

NumPy matrix


For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix for other options.

If the Graph contains selfloops, D is defined as diag(sum(A,1)), where A is the adjacency matrix [2].


[1]Fan Chung-Graham, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, Number 92, 1997.
[2]Steve Butler, Interlacing For Weighted Graphs Using The Normalized Laplacian, Electronic Journal of Linear Algebra, Volume 16, pp. 90-98, March 2007.