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# fiedler_vector

fiedler_vector(G, weight='weight', normalized=False, tol=1e-08, method='tracemin_pcg', seed=None)

  Returns the Fiedler vector of a connected undirected graph.

  The Fiedler vector of a connected undirected graph is the eigenvector corresponding to the second smallest eigenvalue of the Laplacian matrix of the graph.

  Parameters:
    G NetworkX graph

      An undirected graph.

    weight object, optional (default: None)

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

    normalized bool, optional (default: False)

      Whether the normalized Laplacian matrix is used.

    tol float, optional (default: 1e-8)

      Tolerance of relative residual in eigenvalue computation.

    method string, optional (default: ‘tracemin_pcg’)

      Method of eigenvalue computation. It must be one of the tracemin options shown below (TraceMIN), ‘lanczos’ (Lanczos iteration) or ‘lobpcg’ (LOBPCG).

      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_lu’ | LU factorization

    seed integer, random_state, or None (default)

      Indicator of random number generation state. See Randomness.

  Returns:
    fiedler_vector NumPy array of floats.

      Fiedler vector.

  Raises:
    NetworkXNotImplemented

      If G is directed.

    NetworkXError

      If G has less than two nodes or is not connected.

  See also

  `laplacian_matrix`

  Notes

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

  Examples

  Given a connected graph the signs of the values in the Fiedler vector can be used to partition the graph into two components.

      >>> G = nx.barbell_graph(5, 0)
      >>> nx.fiedler_vector(G, normalized=True, seed=1)
      array([-0.32864129, -0.32864129, -0.32864129, -0.32864129, -0.26072899,
              0.26072899,  0.32864129,  0.32864129,  0.32864129,  0.32864129])

  The connected components are the two 5-node cliques of the barbell graph.