Linear algebra

Graph Matrix

Adjacency matrix and incidence matrix of graphs.

adjacency_matrix(G[, nodelist, dtype, weight])	Returns adjacency matrix of G.
incidence_matrix(G[, nodelist, edgelist, ...])	Returns incidence matrix of G.

Laplacian Matrix

Laplacian matrix of graphs.

All calculations here are done using the out-degree. For Laplacians using in-degree, use G.reverse(copy=False) instead of G and take the transpose.

The laplacian_matrix function provides an unnormalized matrix, while normalized_laplacian_matrix, directed_laplacian_matrix, and directed_combinatorial_laplacian_matrix are all normalized.

laplacian_matrix(G[, nodelist, weight])	Returns the Laplacian matrix of G.
normalized_laplacian_matrix(G[, nodelist, ...])	Returns the normalized Laplacian matrix of G.
directed_laplacian_matrix(G[, nodelist, ...])	Returns the directed Laplacian matrix of G.
directed_combinatorial_laplacian_matrix(G[, ...])	Return the directed combinatorial Laplacian matrix of G.
total_spanning_tree_weight(G[, weight, root])	Returns the total weight of all spanning trees of G.

Bethe Hessian Matrix

Bethe Hessian or deformed Laplacian matrix of graphs.

bethe_hessian_matrix(G[, r, nodelist])

Returns the Bethe Hessian matrix of G.

Algebraic Connectivity

Algebraic connectivity and Fiedler vectors of undirected graphs.

algebraic_connectivity(G[, weight, ...])	Returns the algebraic connectivity of an undirected graph.
fiedler_vector(G[, weight, normalized, tol, ...])	Returns the Fiedler vector of a connected undirected graph.
spectral_ordering(G[, weight, normalized, ...])	Compute the spectral_ordering of a graph.
spectral_bisection(G[, weight, normalized, ...])	Bisect the graph using the Fiedler vector.

Attribute Matrices

Functions for constructing matrix-like objects from graph attributes.

attr_matrix(G[, edge_attr, node_attr, ...])	Returns the attribute matrix using attributes from G as a numpy array.
attr_sparse_matrix(G[, edge_attr, ...])	Returns a SciPy sparse array using attributes from G.

Modularity Matrices

Modularity matrix of graphs.

modularity_matrix(G[, nodelist, weight])	Returns the modularity matrix of G.
directed_modularity_matrix(G[, nodelist, weight])	Returns the directed modularity matrix of G.

Spectrum

Eigenvalue spectrum of graphs.

adjacency_spectrum(G[, weight])	Returns eigenvalues of the adjacency matrix of G.
laplacian_spectrum(G[, weight])	Returns eigenvalues of the Laplacian of G
bethe_hessian_spectrum(G[, r])	Returns eigenvalues of the Bethe Hessian matrix of G.
normalized_laplacian_spectrum(G[, weight])	Return eigenvalues of the normalized Laplacian of G
modularity_spectrum(G)	Returns eigenvalues of the modularity matrix of G.