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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.

  • 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_pcg’ Preconditioned conjugate gradient method
    ‘tracemin_chol’ Cholesky factorization
    ‘tracemin_lu’ LU factorization

algebraic_connectivity – Algebraic connectivity.

Return type:


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


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.

See also