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# 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.