networkx.linalg.laplacianmatrix.normalized_laplacian_matrix¶

normalized_laplacian_matrix
(G, nodelist=None, weight='weight')[source]¶ Returns 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 andD
is the diagonal matrix of node degrees. Parameters
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.
 Returns
N – The normalized Laplacian matrix of G.
 Return type
NumPy matrix
Notes
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.
See also
laplacian_matrix()
,normalized_laplacian_spectrum()
References
 1
Fan ChungGraham, 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. 9098, March 2007.