# Source code for networkx.generators.classic

```"""Generators for some classic graphs.

The typical graph builder function is called as follows:

>>> G = nx.complete_graph(100)

returning the complete graph on n nodes labeled 0, .., 99
as a simple graph. Except for `empty_graph`, all the functions
in this module return a Graph class (i.e. a simple, undirected graph).

"""

import itertools
import numbers

import networkx as nx
from networkx.classes import Graph
from networkx.exception import NetworkXError
from networkx.utils import nodes_or_number, pairwise

__all__ = [
"balanced_tree",
"barbell_graph",
"binomial_tree",
"complete_graph",
"complete_multipartite_graph",
"circulant_graph",
"cycle_graph",
"dorogovtsev_goltsev_mendes_graph",
"empty_graph",
"full_rary_tree",
"lollipop_graph",
"null_graph",
"path_graph",
"star_graph",
"trivial_graph",
"turan_graph",
"wheel_graph",
]

# -------------------------------------------------------------------
#   Some Classic Graphs
# -------------------------------------------------------------------

def _tree_edges(n, r):
if n == 0:
return
# helper function for trees
# yields edges in rooted tree at 0 with n nodes and branching ratio r
nodes = iter(range(n))
parents = [next(nodes)]  # stack of max length r
while parents:
source = parents.pop(0)
for i in range(r):
try:
target = next(nodes)
parents.append(target)
yield source, target
except StopIteration:
break

[docs]
def full_rary_tree(r, n, create_using=None):
"""Creates a full r-ary tree of `n` nodes.

Sometimes called a k-ary, n-ary, or m-ary tree.
"... all non-leaf nodes have exactly r children and all levels
are full except for some rightmost position of the bottom level
(if a leaf at the bottom level is missing, then so are all of the
leaves to its right." _

Parameters
----------
r : int
branching factor of the tree
n : int
Number of nodes in the tree
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Returns
-------
G : networkx Graph
An r-ary tree with n nodes

References
----------
..  An introduction to data structures and algorithms,
James Andrew Storer,  Birkhauser Boston 2001, (page 225).
"""
G = empty_graph(n, create_using)
return G

[docs]
def balanced_tree(r, h, create_using=None):
"""Returns the perfectly balanced `r`-ary tree of height `h`.

Parameters
----------
r : int
Branching factor of the tree; each node will have `r`
children.

h : int
Height of the tree.

create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Returns
-------
G : NetworkX graph
A balanced `r`-ary tree of height `h`.

Notes
-----
This is the rooted tree where all leaves are at distance `h` from
the root. The root has degree `r` and all other internal nodes
have degree `r + 1`.

Node labels are integers, starting from zero.

A balanced tree is also known as a *complete r-ary tree*.

"""
# The number of nodes in the balanced tree is `1 + r + ... + r^h`,
# which is computed by using the closed-form formula for a geometric
# sum with ratio `r`. In the special case that `r` is 1, the number
# of nodes is simply `h + 1` (since the tree is actually a path
# graph).
if r == 1:
n = h + 1
else:
# This must be an integer if both `r` and `h` are integers. If
# they are not, we force integer division anyway.
n = (1 - r ** (h + 1)) // (1 - r)
return full_rary_tree(r, n, create_using=create_using)

[docs]
def barbell_graph(m1, m2, create_using=None):
"""Returns the Barbell Graph: two complete graphs connected by a path.

Parameters
----------
m1 : int
Size of the left and right barbells, must be greater than 2.

m2 : int
Length of the path connecting the barbells.

create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Only undirected Graphs are supported.

Returns
-------
G : NetworkX graph
A barbell graph.

Notes
-----

Two identical complete graphs \$K_{m1}\$ form the left and right bells,
and are connected by a path \$P_{m2}\$.

The `2*m1+m2`  nodes are numbered
`0, ..., m1-1` for the left barbell,
`m1, ..., m1+m2-1` for the path,
and `m1+m2, ..., 2*m1+m2-1` for the right barbell.

The 3 subgraphs are joined via the edges `(m1-1, m1)` and
`(m1+m2-1, m1+m2)`. If `m2=0`, this is merely two complete
graphs joined together.

This graph is an extremal example in David Aldous
and Jim Fill's e-text on Random Walks on Graphs.

"""
if m1 < 2:
raise NetworkXError("Invalid graph description, m1 should be >=2")
if m2 < 0:
raise NetworkXError("Invalid graph description, m2 should be >=0")

# left barbell
G = complete_graph(m1, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")

# connecting path
G.add_nodes_from(range(m1, m1 + m2 - 1))
if m2 > 1:

# right barbell
(u, v) for u in range(m1 + m2, 2 * m1 + m2) for v in range(u + 1, 2 * m1 + m2)
)

# connect it up
if m2 > 0:
G.add_edge(m1 + m2 - 1, m1 + m2)

return G

[docs]
def binomial_tree(n, create_using=None):
"""Returns the Binomial Tree of order n.

The binomial tree of order 0 consists of a single node. A binomial tree of order k
is defined recursively by linking two binomial trees of order k-1: the root of one is
the leftmost child of the root of the other.

Parameters
----------
n : int
Order of the binomial tree.

create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Returns
-------
G : NetworkX graph
A binomial tree of \$2^n\$ nodes and \$2^n - 1\$ edges.

"""
G = nx.empty_graph(1, create_using)

N = 1
for i in range(n):
# Use G.edges() to ensure 2-tuples. G.edges is 3-tuple for MultiGraph
edges = [(u + N, v + N) for (u, v) in G.edges()]
N *= 2
return G

[docs]
@nodes_or_number(0)
def complete_graph(n, create_using=None):
"""Return the complete graph `K_n` with n nodes.

A complete graph on `n` nodes means that all pairs
of distinct nodes have an edge connecting them.

Parameters
----------
n : int or iterable container of nodes
If n is an integer, nodes are from range(n).
If n is a container of nodes, those nodes appear in the graph.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Examples
--------
>>> G = nx.complete_graph(9)
>>> len(G)
9
>>> G.size()
36
>>> G = nx.complete_graph(range(11, 14))
>>> list(G.nodes())
[11, 12, 13]
>>> G = nx.complete_graph(4, nx.DiGraph())
>>> G.is_directed()
True

"""
_, nodes = n
G = empty_graph(nodes, create_using)
if len(nodes) > 1:
if G.is_directed():
edges = itertools.permutations(nodes, 2)
else:
edges = itertools.combinations(nodes, 2)
return G

[docs]
"""Returns the circular ladder graph \$CL_n\$ of length n.

\$CL_n\$ consists of two concentric n-cycles in which
each of the n pairs of concentric nodes are joined by an edge.

Node labels are the integers 0 to n-1

"""
G.add_edge(n, 2 * n - 1)
return G

[docs]
def circulant_graph(n, offsets, create_using=None):
r"""Returns the circulant graph \$Ci_n(x_1, x_2, ..., x_m)\$ with \$n\$ nodes.

The circulant graph \$Ci_n(x_1, ..., x_m)\$ consists of \$n\$ nodes \$0, ..., n-1\$
such that node \$i\$ is connected to nodes \$(i + x) \mod n\$ and \$(i - x) \mod n\$
for all \$x\$ in \$x_1, ..., x_m\$. Thus \$Ci_n(1)\$ is a cycle graph.

Parameters
----------
n : integer
The number of nodes in the graph.
offsets : list of integers
A list of node offsets, \$x_1\$ up to \$x_m\$, as described above.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Returns
-------
NetworkX Graph of type create_using

Examples
--------
Many well-known graph families are subfamilies of the circulant graphs;
for example, to create the cycle graph on n points, we connect every
node to nodes on either side (with offset plus or minus one). For n = 10,

>>> G = nx.circulant_graph(10, )
>>> edges = [
...     (0, 9),
...     (0, 1),
...     (1, 2),
...     (2, 3),
...     (3, 4),
...     (4, 5),
...     (5, 6),
...     (6, 7),
...     (7, 8),
...     (8, 9),
... ]
...
>>> sorted(edges) == sorted(G.edges())
True

Similarly, we can create the complete graph
on 5 points with the set of offsets [1, 2]:

>>> G = nx.circulant_graph(5, [1, 2])
>>> edges = [
...     (0, 1),
...     (0, 2),
...     (0, 3),
...     (0, 4),
...     (1, 2),
...     (1, 3),
...     (1, 4),
...     (2, 3),
...     (2, 4),
...     (3, 4),
... ]
...
>>> sorted(edges) == sorted(G.edges())
True

"""
G = empty_graph(n, create_using)
for i in range(n):
for j in offsets:
G.add_edge(i, (i - j) % n)
G.add_edge(i, (i + j) % n)
return G

[docs]
@nodes_or_number(0)
def cycle_graph(n, create_using=None):
"""Returns the cycle graph \$C_n\$ of cyclically connected nodes.

\$C_n\$ is a path with its two end-nodes connected.

Parameters
----------
n : int or iterable container of nodes
If n is an integer, nodes are from `range(n)`.
If n is a container of nodes, those nodes appear in the graph.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Notes
-----
If create_using is directed, the direction is in increasing order.

"""
_, nodes = n
G = empty_graph(nodes, create_using)
return G

[docs]
def dorogovtsev_goltsev_mendes_graph(n, create_using=None):
"""Returns the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph.

The Dorogovtsev-Goltsev-Mendes _ procedure produces a scale-free graph
deterministically with the following properties for a given `n`:
- Total number of nodes = ``3 * (3**n + 1) / 2``
- Total number of edges = ``3 ** (n + 1)``

Parameters
----------
n : integer
The generation number.

create_using : NetworkX Graph, optional
Graph type to be returned. Directed graphs and multi graphs are not
supported.

Returns
-------
G : NetworkX Graph

Examples
--------
>>> G = nx.dorogovtsev_goltsev_mendes_graph(3)
>>> G.number_of_nodes()
15
>>> G.number_of_edges()
27
>>> nx.is_planar(G)
True

References
----------
..  Dorogotsev S.N., Goltsev A.V., and Mendes J.F.F "Pseudofractal
Scale-free Web". arXiv:cond-mat/0112143
"""
G = empty_graph(0, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if G.is_multigraph():
raise NetworkXError("Multigraph not supported")

if n == 0:
return G
new_node = 2  # next node to be added
for i in range(1, n + 1):  # iterate over number of generations.
last_generation_edges = list(G.edges())
number_of_edges_in_last_generation = len(last_generation_edges)
for j in range(0, number_of_edges_in_last_generation):
new_node += 1
return G

[docs]
@nodes_or_number(0)
def empty_graph(n=0, create_using=None, default=Graph):
"""Returns the empty graph with n nodes and zero edges.

Parameters
----------
n : int or iterable container of nodes (default = 0)
If n is an integer, nodes are from `range(n)`.
If n is a container of nodes, those nodes appear in the graph.
create_using : Graph Instance, Constructor or None
Indicator of type of graph to return.
If a Graph-type instance, then clear and use it.
If None, use the `default` constructor.
If a constructor, call it to create an empty graph.
default : Graph constructor (optional, default = nx.Graph)
The constructor to use if create_using is None.
If None, then nx.Graph is used.
This is used when passing an unknown `create_using` value
through your home-grown function to `empty_graph` and
you want a default constructor other than nx.Graph.

Examples
--------
>>> G = nx.empty_graph(10)
>>> G.number_of_nodes()
10
>>> G.number_of_edges()
0
>>> G = nx.empty_graph("ABC")
>>> G.number_of_nodes()
3
>>> sorted(G)
['A', 'B', 'C']

Notes
-----
The variable create_using should be a Graph Constructor or a
"graph"-like object. Constructors, e.g. `nx.Graph` or `nx.MultiGraph`
will be used to create the returned graph. "graph"-like objects
will be cleared (nodes and edges will be removed) and refitted as
an empty "graph" with nodes specified in n. This capability
is useful for specifying the class-nature of the resulting empty
"graph" (i.e. Graph, DiGraph, MyWeirdGraphClass, etc.).

The variable create_using has three main uses:
Firstly, the variable create_using can be used to create an
empty digraph, multigraph, etc.  For example,

>>> n = 10
>>> G = nx.empty_graph(n, create_using=nx.DiGraph)

will create an empty digraph on n nodes.

Secondly, one can pass an existing graph (digraph, multigraph,
etc.) via create_using. For example, if G is an existing graph
(resp. digraph, multigraph, etc.), then empty_graph(n, create_using=G)
will empty G (i.e. delete all nodes and edges using G.clear())
and then add n nodes and zero edges, and return the modified graph.

Thirdly, when constructing your home-grown graph creation function
you can use empty_graph to construct the graph by passing a user
defined create_using to empty_graph. In this case, if you want the
default constructor to be other than nx.Graph, specify `default`.

>>> def mygraph(n, create_using=None):
...     G = nx.empty_graph(n, create_using, nx.MultiGraph)
...     return G
>>> G = mygraph(3)
>>> G.is_multigraph()
True
>>> G = mygraph(3, nx.Graph)
>>> G.is_multigraph()
False

"""
if create_using is None:
G = default()
elif isinstance(create_using, type):
G = create_using()
raise TypeError("create_using is not a valid NetworkX graph type or instance")
else:
# create_using is a NetworkX style Graph
create_using.clear()
G = create_using

_, nodes = n
return G

[docs]
"""Returns the Ladder graph of length n.

This is two paths of n nodes, with
each pair connected by a single edge.

Node labels are the integers 0 to 2*n - 1.

"""
G = empty_graph(2 * n, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
G.add_edges_from((v, v + n) for v in range(n))
return G

[docs]
@nodes_or_number([0, 1])
def lollipop_graph(m, n, create_using=None):
"""Returns the Lollipop Graph; `K_m` connected to `P_n`.

This is the Barbell Graph without the right barbell.

Parameters
----------
m, n : int or iterable container of nodes (default = 0)
If an integer, nodes are from `range(m)` and `range(m,m+n)`.
If a container of nodes, those nodes appear in the graph.
Warning: m and n are not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.

The nodes for m appear in the complete graph \$K_m\$ and the nodes
for n appear in the path \$P_n\$
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Notes
-----
The 2 subgraphs are joined via an edge (m-1, m).
If n=0, this is merely a complete graph.

(This graph is an extremal example in David Aldous and Jim
Fill's etext on Random Walks on Graphs.)

"""
m, m_nodes = m
M = len(m_nodes)
if M < 2:
raise NetworkXError("Invalid description: m should indicate at least 2 nodes")

n, n_nodes = n
if isinstance(m, numbers.Integral) and isinstance(n, numbers.Integral):
n_nodes = list(range(M, M + n))
N = len(n_nodes)

# the ball
G = complete_graph(m_nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")

# the stick
if N > 1:

if len(G) != M + N:
raise NetworkXError("Nodes must be distinct in containers m and n")

# connect ball to stick
if M > 0 and N > 0:
return G

[docs]
def null_graph(create_using=None):
"""Returns the Null graph with no nodes or edges.

See empty_graph for the use of create_using.

"""
G = empty_graph(0, create_using)
return G

[docs]
@nodes_or_number(0)
def path_graph(n, create_using=None):
"""Returns the Path graph `P_n` of linearly connected nodes.

Parameters
----------
n : int or iterable
If an integer, nodes are 0 to n - 1.
If an iterable of nodes, in the order they appear in the path.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

"""
_, nodes = n
G = empty_graph(nodes, create_using)
return G

[docs]
@nodes_or_number(0)
def star_graph(n, create_using=None):
"""Return the star graph

The star graph consists of one center node connected to n outer nodes.

Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(n)  # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")

if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G

[docs]
def trivial_graph(create_using=None):
"""Return the Trivial graph with one node (with label 0) and no edges."""
G = empty_graph(1, create_using)
return G

[docs]
def turan_graph(n, r):
r"""Return the Turan Graph

The Turan Graph is a complete multipartite graph on \$n\$ nodes
with \$r\$ disjoint subsets. That is, edges connect each node to
every node not in its subset.

Given \$n\$ and \$r\$, we create a complete multipartite graph with
\$r-(n \mod r)\$ partitions of size \$n/r\$, rounded down, and
\$n \mod r\$ partitions of size \$n/r+1\$, rounded down.

Parameters
----------
n : int
The number of nodes.
r : int
The number of partitions.
Must be less than or equal to n.

Notes
-----
Must satisfy \$1 <= r <= n\$.
The graph has \$(r-1)(n^2)/(2r)\$ edges, rounded down.
"""

if not 1 <= r <= n:
raise NetworkXError("Must satisfy 1 <= r <= n")

partitions = [n // r] * (r - (n % r)) + [n // r + 1] * (n % r)
G = complete_multipartite_graph(*partitions)
return G

[docs]
@nodes_or_number(0)
def wheel_graph(n, create_using=None):
"""Return the wheel graph

The wheel graph consists of a hub node connected to a cycle of (n-1) nodes.

Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.

Node labels are the integers 0 to n - 1.
"""
_, nodes = n
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")

if len(nodes) > 1:
hub, *rim = nodes
G.add_edges_from((hub, node) for node in rim)
if len(rim) > 1:
return G

[docs]
def complete_multipartite_graph(*subset_sizes):
"""Returns the complete multipartite graph with the specified subset sizes.

Parameters
----------
subset_sizes : tuple of integers or tuple of node iterables
The arguments can either all be integer number of nodes or they
can all be iterables of nodes. If integers, they represent the
number of nodes in each subset of the multipartite graph.
If iterables, each is used to create the nodes for that subset.
The length of subset_sizes is the number of subsets.

Returns
-------
G : NetworkX Graph
Returns the complete multipartite graph with the specified subsets.

For each node, the node attribute 'subset' is an integer
indicating which subset contains the node.

Examples
--------
Creating a complete tripartite graph, with subsets of one, two, and three
nodes, respectively.

>>> G = nx.complete_multipartite_graph(1, 2, 3)
>>> [G.nodes[u]["subset"] for u in G]
[0, 1, 1, 2, 2, 2]
>>> list(G.edges(0))
[(0, 1), (0, 2), (0, 3), (0, 4), (0, 5)]
>>> list(G.edges(2))
[(2, 0), (2, 3), (2, 4), (2, 5)]
>>> list(G.edges(4))
[(4, 0), (4, 1), (4, 2)]

>>> G = nx.complete_multipartite_graph("a", "bc", "def")
>>> [G.nodes[u]["subset"] for u in sorted(G)]
[0, 1, 1, 2, 2, 2]

Notes
-----
This function generalizes several other graph builder functions.

- If no subset sizes are given, this returns the null graph.
- If a single subset size `n` is given, this returns the empty graph on
`n` nodes.
- If two subset sizes `m` and `n` are given, this returns the complete
bipartite graph on `m + n` nodes.
- If subset sizes `1` and `n` are given, this returns the star graph on
`n + 1` nodes.

--------
complete_bipartite_graph
"""
# The complete multipartite graph is an undirected simple graph.
G = Graph()

if len(subset_sizes) == 0:
return G

# set up subsets of nodes
try:
extents = pairwise(itertools.accumulate((0,) + subset_sizes))
subsets = [range(start, end) for start, end in extents]
except TypeError:
subsets = subset_sizes

# add nodes with subset attribute
# while checking that ints are not mixed with iterables
try:
for i, subset in enumerate(subsets):