def Tournament(self,n):
r"""
Returns a tournament on `n` vertices.
In this tournament there is an edge from `i` to `j` if `i<j`.
INPUT:
- ``n`` (integer) -- number of vertices in the tournament.
EXAMPLES::
sage: g = digraphs.Tournament(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()
1
"""
if n<0:
raise ValueError("The number of vertices must be a positive integer.")
g = DiGraph()
g.name("Tournament on "+str(n)+" vertices")
for i in range(n-1):
for j in range(i+1, n):
g.add_edge(i,j)
if n:
from sage.graphs.graph_plot import _circle_embedding
_circle_embedding(g, range(n))
return g
def Circulant(self, n, integers):
r"""
Returns a circulant digraph on `n` vertices from a set of integers.
INPUT:
- ``n`` (integer) -- number of vertices.
- ``integers`` -- the list of integers such that there is an edge from
`i` to `j` if and only if ``(j-i)%n in integers``.
EXAMPLE::
sage: digraphs.Circulant(13,[3,5,7])
Circulant graph ([3, 5, 7]): Digraph on 13 vertices
TESTS::
sage: digraphs.Circulant(13,[3,5,7,"hey"])
Traceback (most recent call last):
...
ValueError: The list must contain only relative integers.
sage: digraphs.Circulant(-2,[3,5,7,3])
Traceback (most recent call last):
...
ValueError: n must be a positive integer
sage: digraphs.Circulant(3,[3,5,7,3.4])
Traceback (most recent call last):
...
ValueError: The list must contain only relative integers.
"""
from sage.graphs.graph_plot import _circle_embedding
from sage.rings.integer_ring import ZZ
# Bad input and loops
loops = False
if not n in ZZ or n <= 0:
raise ValueError("n must be a positive integer")
for i in integers:
if not i in ZZ:
raise ValueError(
"The list must contain only relative integers.")
if (i % n) == 0:
loops = True
G = DiGraph(n,
name="Circulant graph (" + str(integers) + ")",
loops=loops)
_circle_embedding(G, range(n))
for v in range(n):
G.add_edges([(v, (v + j) % n) for j in integers])
return G
def Circulant(self,n,integers):
r"""
Returns a circulant digraph on `n` vertices from a set of integers.
INPUT:
- ``n`` (integer) -- number of vertices.
- ``integers`` -- the list of integers such that there is an edge from
`i` to `j` if and only if ``(j-i)%n in integers``.
EXAMPLE::
sage: digraphs.Circulant(13,[3,5,7])
Circulant graph ([3, 5, 7]): Digraph on 13 vertices
TESTS::
sage: digraphs.Circulant(13,[3,5,7,"hey"])
Traceback (most recent call last):
...
ValueError: The list must contain only relative integers.
sage: digraphs.Circulant(-2,[3,5,7,3])
Traceback (most recent call last):
...
ValueError: n must be a positive integer
sage: digraphs.Circulant(3,[3,5,7,3.4])
Traceback (most recent call last):
...
ValueError: The list must contain only relative integers.
"""
from sage.graphs.graph_plot import _circle_embedding
from sage.rings.integer_ring import ZZ
# Bad input and loops
loops = False
if not n in ZZ or n <= 0:
raise ValueError("n must be a positive integer")
for i in integers:
if not i in ZZ:
raise ValueError("The list must contain only relative integers.")
if (i%n) == 0:
loops = True
G=DiGraph(n, name="Circulant graph ("+str(integers)+")", loops=loops)
_circle_embedding(G, range(n))
for v in range(n):
G.add_edges([(v,(v+j)%n) for j in integers])
return G
def RandomTournament(self, n):
r"""
Returns a random tournament on `n` vertices.
For every pair of vertices, the tournament has an edge from
`i` to `j` with probability `1/2`, otherwise it has an edge
from `j` to `i`.
See :wikipedia:`Tournament_(graph_theory)`
INPUT:
- ``n`` (integer) -- number of vertices.
EXAMPLES::
sage: T = digraphs.RandomTournament(10); T
Random Tournament: Digraph on 10 vertices
sage: T.size() == binomial(10, 2)
True
sage: digraphs.RandomTournament(-1)
Traceback (most recent call last):
...
ValueError: The number of vertices cannot be strictly negative!
"""
from sage.misc.prandom import random
g = DiGraph(n)
g.name("Random Tournament")
for i in range(n - 1):
for j in range(i + 1, n):
if random() <= 0.5:
g.add_edge(i, j)
else:
g.add_edge(j, i)
if n:
from sage.graphs.graph_plot import _circle_embedding
_circle_embedding(g, range(n))
return g
def TransitiveTournament(self, n):
r"""
Returns a transitive tournament on `n` vertices.
In this tournament there is an edge from `i` to `j` if `i<j`.
See :wikipedia:`Tournament_(graph_theory)`
INPUT:
- ``n`` (integer) -- number of vertices in the tournament.
EXAMPLES::
sage: g = digraphs.TransitiveTournament(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()
1
TESTS::
sage: digraphs.TransitiveTournament(-1)
Traceback (most recent call last):
...
ValueError: The number of vertices cannot be strictly negative!
"""
g = DiGraph(n)
g.name("Transitive Tournament")
for i in range(n - 1):
for j in range(i + 1, n):
g.add_edge(i, j)
if n:
from sage.graphs.graph_plot import _circle_embedding
_circle_embedding(g, range(n))
return g
def RandomTournament(self, n):
r"""
Returns a random tournament on `n` vertices.
For every pair of vertices, the tournament has an edge from
`i` to `j` with probability `1/2`, otherwise it has an edge
from `j` to `i`.
See :wikipedia:`Tournament_(graph_theory)`
INPUT:
- ``n`` (integer) -- number of vertices.
EXAMPLES::
sage: T = digraphs.RandomTournament(10); T
Random Tournament: Digraph on 10 vertices
sage: T.size() == binomial(10, 2)
True
sage: digraphs.RandomTournament(-1)
Traceback (most recent call last):
...
ValueError: The number of vertices cannot be strictly negative!
"""
from sage.misc.prandom import random
g = DiGraph(n)
g.name("Random Tournament")
for i in range(n - 1):
for j in range(i + 1, n):
if random() <= .5:
g.add_edge(i, j)
else:
g.add_edge(j, i)
if n:
from sage.graphs.graph_plot import _circle_embedding
_circle_embedding(g, range(n))
return g
def TransitiveTournament(self, n):
r"""
Returns a transitive tournament on `n` vertices.
In this tournament there is an edge from `i` to `j` if `i<j`.
See :wikipedia:`Tournament_(graph_theory)`
INPUT:
- ``n`` (integer) -- number of vertices in the tournament.
EXAMPLES::
sage: g = digraphs.TransitiveTournament(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()
1
TESTS::
sage: digraphs.TransitiveTournament(-1)
Traceback (most recent call last):
...
ValueError: The number of vertices cannot be strictly negative!
"""
g = DiGraph(n)
g.name("Transitive Tournament")
for i in range(n - 1):
for j in range(i + 1, n):
g.add_edge(i, j)
if n:
from sage.graphs.graph_plot import _circle_embedding
_circle_embedding(g, range(n))
return g
def Tournament(self, n):
r"""
Returns a tournament on `n` vertices.
In this tournament there is an edge from `i` to `j` if `i<j`.
INPUT:
- ``n`` (integer) -- number of vertices in the tournament.
EXAMPLES::
sage: g = digraphs.Tournament(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.size()
10
sage: g.automorphism_group().cardinality()
1
"""
if n < 0:
raise ValueError(
"The number of vertices must be a positive integer.")
g = DiGraph()
g.name("Tournament on " + str(n) + " vertices")
for i in range(n - 1):
for j in range(i + 1, n):
g.add_edge(i, j)
if n:
from sage.graphs.graph_plot import _circle_embedding
_circle_embedding(g, range(n))
return g
