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# algebraic_connectivity¶

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.

Value Solver
‘tracemin_chol’ Cholesky factorization
‘tracemin_lu’ LU factorization
Returns:

algebraic_connectivity – Algebraic connectivity.

Return type:

float

Raises:
• NetworkXNotImplemented – If G is directed.
• NetworkXError – If G has less than two nodes.

Notes

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.

laplacian_matrix()