text: """
  Threshold Graphs - Creation, manipulation and identification.
  """
  from math import sqrt
  import networkx as nx
  from networkx.utils import py_random_state
  
  __all__ = ["is_threshold_graph", "find_threshold_graph"]
  
  
  def is_threshold_graph(G):
      """
      Returns `True` if `G` is a threshold graph.
  
      Parameters
      ----------
      G : NetworkX graph instance
          An instance of `Graph`, `DiGraph`, `MultiGraph` or `MultiDiGraph`
  
      Returns
      -------
      bool
          `True` if `G` is a threshold graph, `False` otherwise.
  
      Examples
      --------
      >>> from networkx.algorithms.threshold import is_threshold_graph
      >>> G = nx.path_graph(3)
      >>> is_threshold_graph(G)
      True
      >>> G = nx.barbell_graph(3, 3)
      >>> is_threshold_graph(G)
      False
  
      References
      ----------
      .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph
      """
      return is_threshold_sequence(list(d for n, d in G.degree()))
  
  
  def is_threshold_sequence(degree_sequence):
      """
      Returns True if the sequence is a threshold degree seqeunce.
  
      Uses the property that a threshold graph must be constructed by
      adding either dominating or isolated nodes. Thus, it can be
      deconstructed iteratively by removing a node of degree zero or a
      node that connects to the remaining nodes.  If this deconstruction
      failes then the sequence is not a threshold sequence.
      """
      ds = degree_sequence[:]  # get a copy so we don't destroy original
      ds.sort()
      while ds:
          if ds[0] == 0:  # if isolated node
              ds.pop(0)  # remove it
              continue
          if ds[-1] != len(ds) - 1:  # is the largest degree node dominating?
              return False  # no, not a threshold degree sequence
          ds.pop()  # yes, largest is the dominating node
          ds = [d - 1 for d in ds]  # remove it and decrement all degrees
      return True
  
  
  def creation_sequence(degree_sequence, with_labels=False, compact=False):
      """
      Determines the creation sequence for the given threshold degree sequence.
  
      The creation sequence is a list of single characters 'd'
      or 'i': 'd' for dominating or 'i' for isolated vertices.
      Dominating vertices are connected to all vertices present when it
      is added.  The first node added is by convention 'd'.
      This list can be converted to a string if desired using "".join(cs)
  
      If with_labels==True:
      Returns a list of 2-tuples containing the vertex number
      and a character 'd' or 'i' which describes the type of vertex.
  
      If compact==True:
      Returns the creation sequence in a compact form that is the number
      of 'i's and 'd's alternating.
      Examples:
      [1,2,2,3] represents d,i,i,d,d,i,i,i
      [3,1,2] represents d,d,d,i,d,d
  
      Notice that the first number is the first vertex to be used for
      construction and so is always 'd'.
  
      with_labels and compact cannot both be True.
  
      Returns None if the sequence is not a threshold sequence
      """
      if with_labels and compact:
          raise ValueError("compact sequences cannot be labeled")
  
      # make an indexed copy
      if isinstance(degree_sequence, dict):  # labeled degree seqeunce
          ds = [[degree, label] for (label, degree) in degree_sequence.items()]
      else:
          ds = [[d, i] for i, d in enumerate(degree_sequence)]
      ds.sort()
      cs = []  # creation sequence
      while ds:
          if ds[0][0] == 0:  # isolated node
              (d, v) = ds.pop(0)
              if len(ds) > 0:  # make sure we start with a d
                  cs.insert(0, (v, "i"))
              else:
                  cs.insert(0, (v, "d"))
              continue
          if ds[-1][0] != len(ds) - 1:  # Not dominating node
              return None  # not a threshold degree sequence
          (d, v) = ds.pop()
          cs.insert(0, (v, "d"))
          ds = [[d[0] - 1, d[1]] for d in ds]  # decrement due to removing node
  
      if with_labels:
          return cs
      if compact:
          return make_compact(cs)
      return [v[1] for v in cs]  # not labeled
  
  
  def make_compact(creation_sequence):
      """
      Returns the creation sequence in a compact form
      that is the number of 'i's and 'd's alternating.
  
      Examples
      --------
      >>> from networkx.algorithms.threshold import make_compact
      >>> make_compact(["d", "i", "i", "d", "d", "i", "i", "i"])
      [1, 2, 2, 3]
      >>> make_compact(["d", "d", "d", "i", "d", "d"])
      [3, 1, 2]
  
      Notice that the first number is the first vertex
      to be used for construction and so is always 'd'.
  
      Labeled creation sequences lose their labels in the
      compact representation.
  
      >>> make_compact([3, 1, 2])
      [3, 1, 2]
      """
      first = creation_sequence[0]
      if isinstance(first, str):  # creation sequence
          cs = creation_sequence[:]
      elif isinstance(first, tuple):  # labeled creation sequence
          cs = [s[1] for s in creation_sequence]
      elif isinstance(first, int):  # compact creation sequence
          return creation_sequence
      else:
          raise TypeError("Not a valid creation sequence type")
  
      ccs = []
      count = 1  # count the run lengths of d's or i's.
      for i in range(1, len(cs)):
          if cs[i] == cs[i - 1]:
              count += 1
          else:
              ccs.append(count)
              count = 1
      ccs.append(count)  # don't forget the last one
      return ccs
  
  
  def uncompact(creation_sequence):
      """
      Converts a compact creation sequence for a threshold
      graph to a standard creation sequence (unlabeled).
      If the creation_sequence is already standard, return it.
      See creation_sequence.
      """
      first = creation_sequence[0]
      if isinstance(first, str):  # creation sequence
          return creation_sequence
      elif isinstance(first, tuple):  # labeled creation sequence
          return creation_sequence
      elif isinstance(first, int):  # compact creation sequence
          ccscopy = creation_sequence[:]
      else:
          raise TypeError("Not a valid creation sequence type")
      cs = []
      while ccscopy:
          cs.extend(ccscopy.pop(0) * ["d"])
          if ccscopy:
              cs.extend(ccscopy.pop(0) * ["i"])
      return cs
  
  
  def creation_sequence_to_weights(creation_sequence):
      """
      Returns a list of node weights which create the threshold
      graph designated by the creation sequence.  The weights
      are scaled so that the threshold is 1.0.  The order of the
      nodes is the same as that in the creation sequence.
      """
      # Turn input sequence into a labeled creation sequence
      first = creation_sequence[0]
      if isinstance(first, str):  # creation sequence
          if isinstance(creation_sequence, list):
              wseq = creation_sequence[:]
          else:
              wseq = list(creation_sequence)  # string like 'ddidid'
      elif isinstance(first, tuple):  # labeled creation sequence
          wseq = [v[1] for v in creation_sequence]
      elif isinstance(first, int):  # compact creation sequence
          wseq = uncompact(creation_sequence)
      else:
          raise TypeError("Not a valid creation sequence type")
      # pass through twice--first backwards
      wseq.reverse()
      w = 0
      prev = "i"
      for j, s in enumerate(wseq):
          if s == "i":
              wseq[j] = w
              prev = s
          elif prev == "i":
              prev = s
              w += 1
      wseq.reverse()  # now pass through forwards
      for j, s in enumerate(wseq):
          if s == "d":
              wseq[j] = w
              prev = s
          elif prev == "d":
              prev = s
              w += 1
      # Now scale weights
      if prev == "d":
          w += 1
      wscale = 1.0 / float(w)
      return [ww * wscale for ww in wseq]
      # return wseq
  
  
  def weights_to_creation_sequence(
      weights, threshold=1, with_labels=False, compact=False
  ):
      """
      Returns a creation sequence for a threshold graph
      determined by the weights and threshold given as input.
      If the sum of two node weights is greater than the
      threshold value, an edge is created between these nodes.
  
      The creation sequence is a list of single characters 'd'
      or 'i': 'd' for dominating or 'i' for isolated vertices.
      Dominating vertices are connected to all vertices present
      when it is added.  The first node added is by convention 'd'.
  
      If with_labels==True:
      Returns a list of 2-tuples containing the vertex number
      and a character 'd' or 'i' which describes the type of vertex.
  
      If compact==True:
      Returns the creation sequence in a compact form that is the number
      of 'i's and 'd's alternating.
      Examples:
      [1,2,2,3] represents d,i,i,d,d,i,i,i
      [3,1,2] represents d,d,d,i,d,d
  
      Notice that the first number is the first vertex to be used for
      construction and so is always 'd'.
  
      with_labels and compact cannot both be True.
      """
      if with_labels and compact:
          raise ValueError("compact sequences cannot be labeled")
  
      # make an indexed copy
      if isinstance(weights, dict):  # labeled weights
          wseq = [[w, label] for (label, w) in weights.items()]
      else:
          wseq = [[w, i] for i, w in enumerate(weights)]
      wseq.sort()
      cs = []  # creation sequence
      cutoff = threshold - wseq[-1][0]
      while wseq:
          if wseq[0][0] < cutoff:  # isolated node
              (w, label) = wseq.pop(0)
              cs.append((label, "i"))
          else:
              (w, label) = wseq.pop()
              cs.append((label, "d"))
              cutoff = threshold - wseq[-1][0]
          if len(wseq) == 1:  # make sure we start with a d
              (w, label) = wseq.pop()
              cs.append((label, "d"))
      # put in correct order
      cs.reverse()
  
      if with_labels:
          return cs
      if compact:
          return make_compact(cs)
      return [v[1] for v in cs]  # not labeled
  
  
  # Manipulating NetworkX.Graphs in context of threshold graphs
  def threshold_graph(creation_sequence, create_using=None):
      """
      Create a threshold graph from the creation sequence or compact
      creation_sequence.
  
      The input sequence can be a
  
      creation sequence (e.g. ['d','i','d','d','d','i'])
      labeled creation sequence (e.g. [(0,'d'),(2,'d'),(1,'i')])
      compact creation sequence (e.g. [2,1,1,2,0])
  
      Use cs=creation_sequence(degree_sequence,labeled=True)
      to convert a degree sequence to a creation sequence.
  
      Returns None if the sequence is not valid
      """
      # Turn input sequence into a labeled creation sequence
      first = creation_sequence[0]
      if isinstance(first, str):  # creation sequence
          ci = list(enumerate(creation_sequence))
      elif isinstance(first, tuple):  # labeled creation sequence
          ci = creation_sequence[:]
      elif isinstance(first, int):  # compact creation sequence
          cs = uncompact(creation_sequence)
          ci = list(enumerate(cs))
      else:
          print("not a valid creation sequence type")
          return None
  
      G = nx.empty_graph(0, create_using)
      if G.is_directed():
          raise nx.NetworkXError("Directed Graph not supported")
  
      G.name = "Threshold Graph"
  
      # add nodes and edges
      # if type is 'i' just add nodea
      # if type is a d connect to everything previous
      while ci:
          (v, node_type) = ci.pop(0)
          if node_type == "d":  # dominating type, connect to all existing nodes
              # We use `for u in list(G):` instead of
              # `for u in G:` because we edit the graph `G` in
              # the loop. Hence using an iterator will result in
              # `RuntimeError: dictionary changed size during iteration`
              for u in list(G):
                  G.add_edge(v, u)
          G.add_node(v)
      return G
  
  
  def find_alternating_4_cycle(G):
      """
      Returns False if there aren't any alternating 4 cycles.
      Otherwise returns the cycle as [a,b,c,d] where (a,b)
      and (c,d) are edges and (a,c) and (b,d) are not.
      """
      for (u, v) in G.edges():
          for w in G.nodes():
              if not G.has_edge(u, w) and u != w:
                  for x in G.neighbors(w):
                      if not G.has_edge(v, x) and v != x:
                          return [u, v, w, x]
      return False
  
  
  def find_threshold_graph(G, create_using=None):
      """
      Returns a threshold subgraph that is close to largest in `G`.
  
      The threshold graph will contain the largest degree node in G.
  
      Parameters
      ----------
      G : NetworkX graph instance
          An instance of `Graph`, or `MultiDiGraph`
      create_using : NetworkX graph class or `None` (default), optional
          Type of graph to use when constructing the threshold graph.
          If `None`, infer the appropriate graph type from the input.
  
      Returns
      -------
      graph :
          A graph instance representing the threshold graph
  
      Examples
      --------
      >>> from networkx.algorithms.threshold import find_threshold_graph
      >>> G = nx.barbell_graph(3, 3)
      >>> T = find_threshold_graph(G)
      >>> T.nodes # may vary
      NodeView((7, 8, 5, 6))
  
      References
      ----------
      .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph
      """
      return threshold_graph(find_creation_sequence(G), create_using)
  
  
  def find_creation_sequence(G):
      """
      Find a threshold subgraph that is close to largest in G.
      Returns the labeled creation sequence of that threshold graph.
      """
      cs = []
      # get a local pointer to the working part of the graph
      H = G
      while H.order() > 0:
          # get new degree sequence on subgraph
          dsdict = dict(H.degree())
          ds = [(d, v) for v, d in dsdict.items()]
          ds.sort()
          # Update threshold graph nodes
          if ds[-1][0] == 0:  # all are isolated
              cs.extend(zip(dsdict, ["i"] * (len(ds) - 1) + ["d"]))
              break  # Done!
          # pull off isolated nodes
          while ds[0][0] == 0:
              (d, iso) = ds.pop(0)
              cs.append((iso, "i"))
          # find new biggest node
          (d, bigv) = ds.pop()
          # add edges of star to t_g
          cs.append((bigv, "d"))
          # form subgraph of neighbors of big node
          H = H.subgraph(H.neighbors(bigv))
      cs.reverse()
      return cs
  
  
  # Properties of Threshold Graphs
  def triangles(creation_sequence):
      """
      Compute number of triangles in the threshold graph with the
      given creation sequence.
      """
      # shortcut algorithm that doesn't require computing number
      # of triangles at each node.
      cs = creation_sequence  # alias
      dr = cs.count("d")  # number of d's in sequence
      ntri = dr * (dr - 1) * (dr - 2) / 6  # number of triangles in clique of nd d's
      # now add dr choose 2 triangles for every 'i' in sequence where
      # dr is the number of d's to the right of the current i
      for i, typ in enumerate(cs):
          if typ == "i":
              ntri += dr * (dr - 1) / 2
          else:
              dr -= 1
      return ntri
  
  
  def triangle_sequence(creation_sequence):
      """
      Return triangle sequence for the given threshold graph creation sequence.
  
      """
      cs = creation_sequence
      seq = []
      dr = cs.count("d")  # number of d's to the right of the current pos
      dcur = (dr - 1) * (dr - 2) // 2  # number of triangles through a node of clique dr
      irun = 0  # number of i's in the last run
      drun = 0  # number of d's in the last run
      for i, sym in enumerate(cs):
          if sym == "d":
              drun += 1
              tri = dcur + (dr - 1) * irun  # new triangles at this d
          else:  # cs[i]="i":
              if prevsym == "d":  # new string of i's
                  dcur += (dr - 1) * irun  # accumulate shared shortest paths
                  irun = 0  # reset i run counter
                  dr -= drun  # reduce number of d's to right
                  drun = 0  # reset d run counter
              irun += 1
              tri = dr * (dr - 1) // 2  # new triangles at this i
          seq.append(tri)
          prevsym = sym
      return seq
  
  
  def cluster_sequence(creation_sequence):
      """
      Return cluster sequence for the given threshold graph creation sequence.
      """
      triseq = triangle_sequence(creation_sequence)
      degseq = degree_sequence(creation_sequence)
      cseq = []
      for i, deg in enumerate(degseq):
          tri = triseq[i]
          if deg <= 1:  # isolated vertex or single pair gets cc 0
              cseq.append(0)
              continue
          max_size = (deg * (deg - 1)) // 2
          cseq.append(float(tri) / float(max_size))
      return cseq
  
  
  def degree_sequence(creation_sequence):
      """
      Return degree sequence for the threshold graph with the given
      creation sequence
      """
      cs = creation_sequence  # alias
      seq = []
      rd = cs.count("d")  # number of d to the right
      for i, sym in enumerate(cs):
          if sym == "d":
              rd -= 1
              seq.append(rd + i)
          else:
              seq.append(rd)
      return seq
  
  
  def density(creation_sequence):
      """
      Return the density of the graph with this creation_sequence.
      The density is the fraction of possible edges present.
      """
      N = len(creation_sequence)
      two_size = sum(degree_sequence(creation_sequence))
      two_possible = N * (N - 1)
      den = two_size / float(two_possible)
      return den
  
  
  def degree_correlation(creation_sequence):
      """
      Return the degree-degree correlation over all edges.
      """
      cs = creation_sequence
      s1 = 0  # deg_i*deg_j
      s2 = 0  # deg_i^2+deg_j^2
      s3 = 0  # deg_i+deg_j
      m = 0  # number of edges
      rd = cs.count("d")  # number of d nodes to the right
      rdi = [i for i, sym in enumerate(cs) if sym == "d"]  # index of "d"s
      ds = degree_sequence(cs)
      for i, sym in enumerate(cs):
          if sym == "d":
              if i != rdi[0]:
                  print("Logic error in degree_correlation", i, rdi)
                  raise ValueError
              rdi.pop(0)
          degi = ds[i]
          for dj in rdi:
              degj = ds[dj]
              s1 += degj * degi
              s2 += degi ** 2 + degj ** 2
              s3 += degi + degj
              m += 1
      denom = 2 * m * s2 - s3 * s3
      numer = 4 * m * s1 - s3 * s3
      if denom == 0:
          if numer == 0:
              return 1
          raise ValueError(f"Zero Denominator but Numerator is {numer}")
      return numer / float(denom)
  
  
  def shortest_path(creation_sequence, u, v):
      """
      Find the shortest path between u and v in a
      threshold graph G with the given creation_sequence.
  
      For an unlabeled creation_sequence, the vertices
      u and v must be integers in (0,len(sequence)) referring
      to the position of the desired vertices in the sequence.
  
      For a labeled creation_sequence, u and v are labels of veritices.
  
      Use cs=creation_sequence(degree_sequence,with_labels=True)
      to convert a degree sequence to a creation sequence.
  
      Returns a list of vertices from u to v.
      Example: if they are neighbors, it returns [u,v]
      """
      # Turn input sequence into a labeled creation sequence
      first = creation_sequence[0]
      if isinstance(first, str):  # creation sequence
          cs = [(i, creation_sequence[i]) for i in range(len(creation_sequence))]
      elif isinstance(first, tuple):  # labeled creation sequence
          cs = creation_sequence[:]
      elif isinstance(first, int):  # compact creation sequence
          ci = uncompact(creation_sequence)
          cs = [(i, ci[i]) for i in range(len(ci))]
      else:
          raise TypeError("Not a valid creation sequence type")
  
      verts = [s[0] for s in cs]
      if v not in verts:
          raise ValueError(f"Vertex {v} not in graph from creation_sequence")
      if u not in verts:
          raise ValueError(f"Vertex {u} not in graph from creation_sequence")
      # Done checking
      if u == v:
          return [u]
  
      uindex = verts.index(u)
      vindex = verts.index(v)
      bigind = max(uindex, vindex)
      if cs[bigind][1] == "d":
          return [u, v]
      # must be that cs[bigind][1]=='i'
      cs = cs[bigind:]
      while cs:
          vert = cs.pop()
          if vert[1] == "d":
              return [u, vert[0], v]
      # All after u are type 'i' so no connection
      return -1
  
  
  def shortest_path_length(creation_sequence, i):
      """
      Return the shortest path length from indicated node to
      every other node for the threshold graph with the given
      creation sequence.
      Node is indicated by index i in creation_sequence unless
      creation_sequence is labeled in which case, i is taken to
      be the label of the node.
  
      Paths lengths in threshold graphs are at most 2.
      Length to unreachable nodes is set to -1.
      """
      # Turn input sequence into a labeled creation sequence
      first = creation_sequence[0]
      if isinstance(first, str):  # creation sequence
          if isinstance(creation_sequence, list):
              cs = creation_sequence[:]
          else:
              cs = list(creation_sequence)
      elif isinstance(first, tuple):  # labeled creation sequence
          cs = [v[1] for v in creation_sequence]
          i = [v[0] for v in creation_sequence].index(i)
      elif isinstance(first, int):  # compact creation sequence
          cs = uncompact(creation_sequence)
      else:
          raise TypeError("Not a valid creation sequence type")
  
      # Compute
      N = len(cs)
      spl = [2] * N  # length 2 to every node
      spl[i] = 0  # except self which is 0
      # 1 for all d's to the right
      for j in range(i + 1, N):
          if cs[j] == "d":
              spl[j] = 1
      if cs[i] == "d":  # 1 for all nodes to the left
          for j in range(i):
              spl[j] = 1
      # and -1 for any trailing i to indicate unreachable
      for j in range(N - 1, 0, -1):
          if cs[j] == "d":
              break
          spl[j] = -1
      return spl
  
  
  def betweenness_sequence(creation_sequence, normalized=True):
      """
      Return betweenness for the threshold graph with the given creation
      sequence.  The result is unscaled.  To scale the values
      to the iterval [0,1] divide by (n-1)*(n-2).
      """
      cs = creation_sequence
      seq = []  # betweenness
      lastchar = "d"  # first node is always a 'd'
      dr = float(cs.count("d"))  # number of d's to the right of curren pos
      irun = 0  # number of i's in the last run
      drun = 0  # number of d's in the last run
      dlast = 0.0  # betweenness of last d
      for i, c in enumerate(cs):
          if c == "d":  # cs[i]=="d":
              # betweennees = amt shared with eariler d's and i's
              #             + new isolated nodes covered
              #             + new paths to all previous nodes
              b = dlast + (irun - 1) * irun / dr + 2 * irun * (i - drun - irun) / dr
              drun += 1  # update counter
          else:  # cs[i]="i":
              if lastchar == "d":  # if this is a new run of i's
                  dlast = b  # accumulate betweenness
                  dr -= drun  # update number of d's to the right
                  drun = 0  # reset d counter
                  irun = 0  # reset i counter
              b = 0  # isolated nodes have zero betweenness
              irun += 1  # add another i to the run
          seq.append(float(b))
          lastchar = c
  
      # normalize by the number of possible shortest paths
      if normalized:
          order = len(cs)
          scale = 1.0 / ((order - 1) * (order - 2))
          seq = [s * scale for s in seq]
  
      return seq
  
  
  def eigenvectors(creation_sequence):
      """
      Return a 2-tuple of Laplacian eigenvalues and eigenvectors
      for the threshold network with creation_sequence.
      The first value is a list of eigenvalues.
      The second value is a list of eigenvectors.
      The lists are in the same order so corresponding eigenvectors
      and eigenvalues are in the same position in the two lists.
  
      Notice that the order of the eigenvalues returned by eigenvalues(cs)
      may not correspond to the order of these eigenvectors.
      """
      ccs = make_compact(creation_sequence)
      N = sum(ccs)
      vec = [0] * N
      val = vec[:]
      # get number of type d nodes to the right (all for first node)
      dr = sum(ccs[::2])
  
      nn = ccs[0]
      vec[0] = [1.0 / sqrt(N)] * N
      val[0] = 0
      e = dr
      dr -= nn
      type_d = True
      i = 1
      dd = 1
      while dd < nn:
          scale = 1.0 / sqrt(dd * dd + i)
          vec[i] = i * [-scale] + [dd * scale] + [0] * (N - i - 1)
          val[i] = e
          i += 1
          dd += 1
      if len(ccs) == 1:
          return (val, vec)
      for nn in ccs[1:]:
          scale = 1.0 / sqrt(nn * i * (i + nn))
          vec[i] = i * [-nn * scale] + nn * [i * scale] + [0] * (N - i - nn)
          # find eigenvalue
          type_d = not type_d
          if type_d:
              e = i + dr
              dr -= nn
          else:
              e = dr
          val[i] = e
          st = i
          i += 1
          dd = 1
          while dd < nn:
              scale = 1.0 / sqrt(i - st + dd * dd)
              vec[i] = [0] * st + (i - st) * [-scale] + [dd * scale] + [0] * (N - i - 1)
              val[i] = e
              i += 1
              dd += 1
      return (val, vec)
  
  
  def spectral_projection(u, eigenpairs):
      """
      Returns the coefficients of each eigenvector
      in a projection of the vector u onto the normalized
      eigenvectors which are contained in eigenpairs.
  
      eigenpairs should be a list of two objects.  The
      first is a list of eigenvalues and the second a list
      of eigenvectors.  The eigenvectors should be lists.
  
      There's not a lot of error checking on lengths of
      arrays, etc. so be careful.
      """
      coeff = []
      evect = eigenpairs[1]
      for ev in evect:
          c = sum([evv * uv for (evv, uv) in zip(ev, u)])
          coeff.append(c)
      return coeff
  
  
  def eigenvalues(creation_sequence):
      """
      Return sequence of eigenvalues of the Laplacian of the threshold
      graph for the given creation_sequence.
  
      Based on the Ferrer's diagram method.  The spectrum is integral
      and is the conjugate of the degree sequence.
  
      See::
  
        @Article{degree-merris-1994,
         author = {Russel Merris},
         title = {Degree maximal graphs are Laplacian integral},
         journal = {Linear Algebra Appl.},
         year = {1994},
         volume = {199},
         pages = {381--389},
        }
  
      """
      degseq = degree_sequence(creation_sequence)
      degseq.sort()
      eiglist = []  # zero is always one eigenvalue
      eig = 0
      row = len(degseq)
      bigdeg = degseq.pop()
      while row:
          if bigdeg < row:
              eiglist.append(eig)
              row -= 1
          else:
              eig += 1
              if degseq:
                  bigdeg = degseq.pop()
              else:
                  bigdeg = 0
      return eiglist
  
  
  # Threshold graph creation routines
  
  
  @py_random_state(2)
  def random_threshold_sequence(n, p, seed=None):
      """
      Create a random threshold sequence of size n.
      A creation sequence is built by randomly choosing d's with
      probabiliy p and i's with probability 1-p.
  
      s=nx.random_threshold_sequence(10,0.5)
  
      returns a threshold sequence of length 10 with equal
      probably of an i or a d at each position.
  
      A "random" threshold graph can be built with
  
      G=nx.threshold_graph(s)
  
      seed : integer, random_state, or None (default)
          Indicator of random number generation state.
          See :ref:`Randomness<randomness>`.
      """
      if not (0 <= p <= 1):
          raise ValueError("p must be in [0,1]")
  
      cs = ["d"]  # threshold sequences always start with a d
      for i in range(1, n):
          if seed.random() < p:
              cs.append("d")
          else:
              cs.append("i")
      return cs
  
  
  # maybe *_d_threshold_sequence routines should
  # be (or be called from) a single routine with a more descriptive name
  # and a keyword parameter?
  def right_d_threshold_sequence(n, m):
      """
      Create a skewed threshold graph with a given number
      of vertices (n) and a given number of edges (m).
  
      The routine returns an unlabeled creation sequence
      for the threshold graph.
  
      FIXME: describe algorithm
  
      """
      cs = ["d"] + ["i"] * (n - 1)  # create sequence with n insolated nodes
  
      #  m <n : not enough edges, make disconnected
      if m < n:
          cs[m] = "d"
          return cs
  
      # too many edges
      if m > n * (n - 1) / 2:
          raise ValueError("Too many edges for this many nodes.")
  
      # connected case m >n-1
      ind = n - 1
      sum = n - 1
      while sum < m:
          cs[ind] = "d"
          ind -= 1
          sum += ind
      ind = m - (sum - ind)
      cs[ind] = "d"
      return cs
  
  
  def left_d_threshold_sequence(n, m):
      """
      Create a skewed threshold graph with a given number
      of vertices (n) and a given number of edges (m).
  
      The routine returns an unlabeled creation sequence
      for the threshold graph.
  
      FIXME: describe algorithm
  
      """
      cs = ["d"] + ["i"] * (n - 1)  # create sequence with n insolated nodes
  
      #  m <n : not enough edges, make disconnected
      if m < n:
          cs[m] = "d"
          return cs
  
      # too many edges
      if m > n * (n - 1) / 2:
          raise ValueError("Too many edges for this many nodes.")
  
      # Connected case when M>N-1
      cs[n - 1] = "d"
      sum = n - 1
      ind = 1
      while sum < m:
          cs[ind] = "d"
          sum += ind
          ind += 1
      if sum > m:  # be sure not to change the first vertex
          cs[sum - m] = "i"
      return cs
  
  
  @py_random_state(3)
  def swap_d(cs, p_split=1.0, p_combine=1.0, seed=None):
      """
      Perform a "swap" operation on a threshold sequence.
  
      The swap preserves the number of nodes and edges
      in the graph for the given sequence.
      The resulting sequence is still a threshold sequence.
  
      Perform one split and one combine operation on the
      'd's of a creation sequence for a threshold graph.
      This operation maintains the number of nodes and edges
      in the graph, but shifts the edges from node to node
      maintaining the threshold quality of the graph.
  
      seed : integer, random_state, or None (default)
          Indicator of random number generation state.
          See :ref:`Randomness<randomness>`.
      """
      # preprocess the creation sequence
      dlist = [i for (i, node_type) in enumerate(cs[1:-1]) if node_type == "d"]
      # split
      if seed.random() < p_split:
          choice = seed.choice(dlist)
          split_to = seed.choice(range(choice))
          flip_side = choice - split_to
          if split_to != flip_side and cs[split_to] == "i" and cs[flip_side] == "i":
              cs[choice] = "i"
              cs[split_to] = "d"
              cs[flip_side] = "d"
              dlist.remove(choice)
              # don't add or combine may reverse this action
              # dlist.extend([split_to,flip_side])
      #            print >>sys.stderr,"split at %s to %s and %s"%(choice,split_to,flip_side)
      # combine
      if seed.random() < p_combine and dlist:
          first_choice = seed.choice(dlist)
          second_choice = seed.choice(dlist)
          target = first_choice + second_choice
          if target >= len(cs) or cs[target] == "d" or first_choice == second_choice:
              return cs
          # OK to combine
          cs[first_choice] = "i"
          cs[second_choice] = "i"
          cs[target] = "d"
      #        print >>sys.stderr,"combine %s and %s to make %s."%(first_choice,second_choice,target)
  
      return cs
  