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algebraic_connectivity(G, weight='weight', normalized=False, tol=1e-08, method='tracemin')[source]

Return the algebraic connectivity of an undirected graph.

The algebraic connectivity of a connected undirected graph is the second smallest eigenvalue of its Laplacian matrix.

Parameters :

G : NetworkX graph

An undirected graph.

weight : object, optional

The data key used to determine the weight of each edge. If None, then each edge has unit weight. Default value: None.

normalized : bool, optional

Whether the normalized Laplacian matrix is used. Default value: False.

tol : float, optional

Tolerance of relative residual in eigenvalue computation. Default value: 1e-8.

method : string, optional

Method of eigenvalue computation. It should be one of ‘tracemin’ (TraceMIN), ‘lanczos’ (Lanczos iteration) and ‘lobpcg’ (LOBPCG). Default value: ‘tracemin’.

The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used.




Preconditioned conjugate gradient method


Cholesky factorization


LU factorization

Returns :

algebraic_connectivity : float

Algebraic connectivity.

Raises :


If G is directed.


If G has less than two nodes.

See also



Edge weights are interpreted by their absolute values. For MultiGraph’s, weights of parallel edges are summed. Zero-weighted edges are ignored.

To use Cholesky factorization in the TraceMIN algorithm, the scikits.sparse package must be installed.