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+"""
+Algebraic connectivity and Fiedler vectors of undirected graphs.
+"""
+
+from functools import partial
+
+import networkx as nx
+from networkx.utils import (
+    not_implemented_for,
+    np_random_state,
+    reverse_cuthill_mckee_ordering,
+)
+
+__all__ = [
+    "algebraic_connectivity",
+    "fiedler_vector",
+    "spectral_ordering",
+    "spectral_bisection",
+]
+
+
+class _PCGSolver:
+    """Preconditioned conjugate gradient method.
+
+    To solve Ax = b:
+        M = A.diagonal() # or some other preconditioner
+        solver = _PCGSolver(lambda x: A * x, lambda x: M * x)
+        x = solver.solve(b)
+
+    The inputs A and M are functions which compute
+    matrix multiplication on the argument.
+    A - multiply by the matrix A in Ax=b
+    M - multiply by M, the preconditioner surrogate for A
+
+    Warning: There is no limit on number of iterations.
+    """
+
+    def __init__(self, A, M):
+        self._A = A
+        self._M = M
+
+    def solve(self, B, tol):
+        import numpy as np
+
+        # Densifying step - can this be kept sparse?
+        B = np.asarray(B)
+        X = np.ndarray(B.shape, order="F")
+        for j in range(B.shape[1]):
+            X[:, j] = self._solve(B[:, j], tol)
+        return X
+
+    def _solve(self, b, tol):
+        import numpy as np
+        import scipy as sp
+
+        A = self._A
+        M = self._M
+        tol *= sp.linalg.blas.dasum(b)
+        # Initialize.
+        x = np.zeros(b.shape)
+        r = b.copy()
+        z = M(r)
+        rz = sp.linalg.blas.ddot(r, z)
+        p = z.copy()
+        # Iterate.
+        while True:
+            Ap = A(p)
+            alpha = rz / sp.linalg.blas.ddot(p, Ap)
+            x = sp.linalg.blas.daxpy(p, x, a=alpha)
+            r = sp.linalg.blas.daxpy(Ap, r, a=-alpha)
+            if sp.linalg.blas.dasum(r) < tol:
+                return x
+            z = M(r)
+            beta = sp.linalg.blas.ddot(r, z)
+            beta, rz = beta / rz, beta
+            p = sp.linalg.blas.daxpy(p, z, a=beta)
+
+
+class _LUSolver:
+    """LU factorization.
+
+    To solve Ax = b:
+        solver = _LUSolver(A)
+        x = solver.solve(b)
+
+    optional argument `tol` on solve method is ignored but included
+    to match _PCGsolver API.
+    """
+
+    def __init__(self, A):
+        import scipy as sp
+
+        self._LU = sp.sparse.linalg.splu(
+            A,
+            permc_spec="MMD_AT_PLUS_A",
+            diag_pivot_thresh=0.0,
+            options={"Equil": True, "SymmetricMode": True},
+        )
+
+    def solve(self, B, tol=None):
+        import numpy as np
+
+        B = np.asarray(B)
+        X = np.ndarray(B.shape, order="F")
+        for j in range(B.shape[1]):
+            X[:, j] = self._LU.solve(B[:, j])
+        return X
+
+
+def _preprocess_graph(G, weight):
+    """Compute edge weights and eliminate zero-weight edges."""
+    if G.is_directed():
+        H = nx.MultiGraph()
+        H.add_nodes_from(G)
+        H.add_weighted_edges_from(
+            ((u, v, e.get(weight, 1.0)) for u, v, e in G.edges(data=True) if u != v),
+            weight=weight,
+        )
+        G = H
+    if not G.is_multigraph():
+        edges = (
+            (u, v, abs(e.get(weight, 1.0))) for u, v, e in G.edges(data=True) if u != v
+        )
+    else:
+        edges = (
+            (u, v, sum(abs(e.get(weight, 1.0)) for e in G[u][v].values()))
+            for u, v in G.edges()
+            if u != v
+        )
+    H = nx.Graph()
+    H.add_nodes_from(G)
+    H.add_weighted_edges_from((u, v, e) for u, v, e in edges if e != 0)
+    return H
+
+
+def _rcm_estimate(G, nodelist):
+    """Estimate the Fiedler vector using the reverse Cuthill-McKee ordering."""
+    import numpy as np
+
+    G = G.subgraph(nodelist)
+    order = reverse_cuthill_mckee_ordering(G)
+    n = len(nodelist)
+    index = dict(zip(nodelist, range(n)))
+    x = np.ndarray(n, dtype=float)
+    for i, u in enumerate(order):
+        x[index[u]] = i
+    x -= (n - 1) / 2.0
+    return x
+
+
+def _tracemin_fiedler(L, X, normalized, tol, method):
+    """Compute the Fiedler vector of L using the TraceMIN-Fiedler algorithm.
+
+    The Fiedler vector of a connected undirected graph is the eigenvector
+    corresponding to the second smallest eigenvalue of the Laplacian matrix
+    of the graph. This function starts with the Laplacian L, not the Graph.
+
+    Parameters
+    ----------
+    L : Laplacian of a possibly weighted or normalized, but undirected graph
+
+    X : Initial guess for a solution. Usually a matrix of random numbers.
+        This function allows more than one column in X to identify more than
+        one eigenvector if desired.
+
+    normalized : bool
+        Whether the normalized Laplacian matrix is used.
+
+    tol : float
+        Tolerance of relative residual in eigenvalue computation.
+        Warning: There is no limit on number of iterations.
+
+    method : string
+        Should be 'tracemin_pcg' or 'tracemin_lu'.
+        Otherwise exception is raised.
+
+    Returns
+    -------
+    sigma, X : Two NumPy arrays of floats.
+        The lowest eigenvalues and corresponding eigenvectors of L.
+        The size of input X determines the size of these outputs.
+        As this is for Fiedler vectors, the zero eigenvalue (and
+        constant eigenvector) are avoided.
+    """
+    import numpy as np
+    import scipy as sp
+
+    n = X.shape[0]
+
+    if normalized:
+        # Form the normalized Laplacian matrix and determine the eigenvector of
+        # its nullspace.
+        e = np.sqrt(L.diagonal())
+        # TODO: rm csr_array wrapper when spdiags array creation becomes available
+        D = sp.sparse.csr_array(sp.sparse.spdiags(1 / e, 0, n, n, format="csr"))
+        L = D @ L @ D
+        e *= 1.0 / np.linalg.norm(e, 2)
+
+    if normalized:
+
+        def project(X):
+            """Make X orthogonal to the nullspace of L."""
+            X = np.asarray(X)
+            for j in range(X.shape[1]):
+                X[:, j] -= (X[:, j] @ e) * e
+
+    else:
+
+        def project(X):
+            """Make X orthogonal to the nullspace of L."""
+            X = np.asarray(X)
+            for j in range(X.shape[1]):
+                X[:, j] -= X[:, j].sum() / n
+
+    if method == "tracemin_pcg":
+        D = L.diagonal().astype(float)
+        solver = _PCGSolver(lambda x: L @ x, lambda x: D * x)
+    elif method == "tracemin_lu":
+        # Convert A to CSC to suppress SparseEfficiencyWarning.
+        A = sp.sparse.csc_array(L, dtype=float, copy=True)
+        # Force A to be nonsingular. Since A is the Laplacian matrix of a
+        # connected graph, its rank deficiency is one, and thus one diagonal
+        # element needs to modified. Changing to infinity forces a zero in the
+        # corresponding element in the solution.
+        i = (A.indptr[1:] - A.indptr[:-1]).argmax()
+        A[i, i] = np.inf
+        solver = _LUSolver(A)
+    else:
+        raise nx.NetworkXError(f"Unknown linear system solver: {method}")
+
+    # Initialize.
+    Lnorm = abs(L).sum(axis=1).flatten().max()
+    project(X)
+    W = np.ndarray(X.shape, order="F")
+
+    while True:
+        # Orthonormalize X.
+        X = np.linalg.qr(X)[0]
+        # Compute iteration matrix H.
+        W[:, :] = L @ X
+        H = X.T @ W
+        sigma, Y = sp.linalg.eigh(H, overwrite_a=True)
+        # Compute the Ritz vectors.
+        X = X @ Y
+        # Test for convergence exploiting the fact that L * X == W * Y.
+        res = sp.linalg.blas.dasum(W @ Y[:, 0] - sigma[0] * X[:, 0]) / Lnorm
+        if res < tol:
+            break
+        # Compute X = L \ X / (X' * (L \ X)).
+        # L \ X can have an arbitrary projection on the nullspace of L,
+        # which will be eliminated.
+        W[:, :] = solver.solve(X, tol)
+        X = (sp.linalg.inv(W.T @ X) @ W.T).T  # Preserves Fortran storage order.
+        project(X)
+
+    return sigma, np.asarray(X)
+
+
+def _get_fiedler_func(method):
+    """Returns a function that solves the Fiedler eigenvalue problem."""
+    import numpy as np
+
+    if method == "tracemin":  # old style keyword <v2.1
+        method = "tracemin_pcg"
+    if method in ("tracemin_pcg", "tracemin_lu"):
+
+        def find_fiedler(L, x, normalized, tol, seed):
+            q = 1 if method == "tracemin_pcg" else min(4, L.shape[0] - 1)
+            X = np.asarray(seed.normal(size=(q, L.shape[0]))).T
+            sigma, X = _tracemin_fiedler(L, X, normalized, tol, method)
+            return sigma[0], X[:, 0]
+
+    elif method == "lanczos" or method == "lobpcg":
+
+        def find_fiedler(L, x, normalized, tol, seed):
+            import scipy as sp
+
+            L = sp.sparse.csc_array(L, dtype=float)
+            n = L.shape[0]
+            if normalized:
+                # TODO: rm csc_array wrapping when spdiags array becomes available
+                D = sp.sparse.csc_array(
+                    sp.sparse.spdiags(
+                        1.0 / np.sqrt(L.diagonal()), [0], n, n, format="csc"
+                    )
+                )
+                L = D @ L @ D
+            if method == "lanczos" or n < 10:
+                # Avoid LOBPCG when n < 10 due to
+                # https://github.com/scipy/scipy/issues/3592
+                # https://github.com/scipy/scipy/pull/3594
+                sigma, X = sp.sparse.linalg.eigsh(
+                    L, 2, which="SM", tol=tol, return_eigenvectors=True
+                )
+                return sigma[1], X[:, 1]
+            else:
+                X = np.asarray(np.atleast_2d(x).T)
+                # TODO: rm csr_array wrapping when spdiags array becomes available
+                M = sp.sparse.csr_array(sp.sparse.spdiags(1.0 / L.diagonal(), 0, n, n))
+                Y = np.ones(n)
+                if normalized:
+                    Y /= D.diagonal()
+                sigma, X = sp.sparse.linalg.lobpcg(
+                    L, X, M=M, Y=np.atleast_2d(Y).T, tol=tol, maxiter=n, largest=False
+                )
+                return sigma[0], X[:, 0]
+
+    else:
+        raise nx.NetworkXError(f"unknown method {method!r}.")
+
+    return find_fiedler
+
+
+@not_implemented_for("directed")
+@np_random_state(5)
+@nx._dispatchable(edge_attrs="weight")
+def algebraic_connectivity(
+    G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
+):
+    r"""Returns 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 (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 :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    algebraic_connectivity : float
+        Algebraic connectivity.
+
+    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.
+
+    See Also
+    --------
+    laplacian_matrix
+
+    Examples
+    --------
+    For undirected graphs algebraic connectivity can tell us if a graph is connected or not
+    `G` is connected iff  ``algebraic_connectivity(G) > 0``:
+
+    >>> G = nx.complete_graph(5)
+    >>> nx.algebraic_connectivity(G) > 0
+    True
+    >>> G.add_node(10)  # G is no longer connected
+    >>> nx.algebraic_connectivity(G) > 0
+    False
+
+    """
+    if len(G) < 2:
+        raise nx.NetworkXError("graph has less than two nodes.")
+    G = _preprocess_graph(G, weight)
+    if not nx.is_connected(G):
+        return 0.0
+
+    L = nx.laplacian_matrix(G)
+    if L.shape[0] == 2:
+        return 2.0 * float(L[0, 0]) if not normalized else 2.0
+
+    find_fiedler = _get_fiedler_func(method)
+    x = None if method != "lobpcg" else _rcm_estimate(G, G)
+    sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
+    return float(sigma)
+
+
+@not_implemented_for("directed")
+@np_random_state(5)
+@nx._dispatchable(edge_attrs="weight")
+def fiedler_vector(
+    G, weight="weight", normalized=False, tol=1e-8, 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 :ref:`Randomness<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.
+
+    Notes
+    -----
+    Edge weights are interpreted by their absolute values. For MultiGraph's,
+    weights of parallel edges are summed. Zero-weighted edges are ignored.
+
+    See Also
+    --------
+    laplacian_matrix
+
+    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.
+    """
+    import numpy as np
+
+    if len(G) < 2:
+        raise nx.NetworkXError("graph has less than two nodes.")
+    G = _preprocess_graph(G, weight)
+    if not nx.is_connected(G):
+        raise nx.NetworkXError("graph is not connected.")
+
+    if len(G) == 2:
+        return np.array([1.0, -1.0])
+
+    find_fiedler = _get_fiedler_func(method)
+    L = nx.laplacian_matrix(G)
+    x = None if method != "lobpcg" else _rcm_estimate(G, G)
+    sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
+    return fiedler
+
+
+@np_random_state(5)
+@nx._dispatchable(edge_attrs="weight")
+def spectral_ordering(
+    G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
+):
+    """Compute the spectral_ordering of a graph.
+
+    The spectral ordering of a graph is an ordering of its nodes where nodes
+    in the same weakly connected components appear contiguous and ordered by
+    their corresponding elements in the Fiedler vector of the component.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        A 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 :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    spectral_ordering : NumPy array of floats.
+        Spectral ordering of nodes.
+
+    Raises
+    ------
+    NetworkXError
+        If G is empty.
+
+    Notes
+    -----
+    Edge weights are interpreted by their absolute values. For MultiGraph's,
+    weights of parallel edges are summed. Zero-weighted edges are ignored.
+
+    See Also
+    --------
+    laplacian_matrix
+    """
+    if len(G) == 0:
+        raise nx.NetworkXError("graph is empty.")
+    G = _preprocess_graph(G, weight)
+
+    find_fiedler = _get_fiedler_func(method)
+    order = []
+    for component in nx.connected_components(G):
+        size = len(component)
+        if size > 2:
+            L = nx.laplacian_matrix(G, component)
+            x = None if method != "lobpcg" else _rcm_estimate(G, component)
+            sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
+            sort_info = zip(fiedler, range(size), component)
+            order.extend(u for x, c, u in sorted(sort_info))
+        else:
+            order.extend(component)
+
+    return order
+
+
+@nx._dispatchable(edge_attrs="weight")
+def spectral_bisection(
+    G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
+):
+    """Bisect the graph using the Fiedler vector.
+
+    This method uses the Fiedler vector to bisect a graph.
+    The partition is defined by the nodes which are associated with
+    either positive or negative values in the vector.
+
+    Parameters
+    ----------
+    G : NetworkX Graph
+
+    weight : str, optional (default: weight)
+        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 :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    bisection : tuple of sets
+        Sets with the bisection of nodes
+
+    Examples
+    --------
+    >>> G = nx.barbell_graph(3, 0)
+    >>> nx.spectral_bisection(G)
+    ({0, 1, 2}, {3, 4, 5})
+
+    References
+    ----------
+    .. [1] M. E. J Newman 'Networks: An Introduction', pages 364-370
+       Oxford University Press 2011.
+    """
+    import numpy as np
+
+    v = nx.fiedler_vector(G, weight, normalized, tol, method, seed)
+    nodes = np.array(list(G))
+    pos_vals = v >= 0
+
+    return set(nodes[~pos_vals].tolist()), set(nodes[pos_vals].tolist())