# 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. 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 Returns eigenvalues of the modularity matrix of G.