def TaylorTwographSRG(q):
r"""
constructing a strongly regular graph from the Taylor's two-graph for `U_3(q)`, `q` odd
This is a strongly regular graph with parameters
`(v,k,\lambda,\mu)=(q^3+1, q(q^2+1)/2, (q^2+3)(q-1)/4, (q^2+1)(q+1)/4)`
in the Seidel switching class of
:func:`Taylor two-graph <sage.combinat.designs.twographs.taylor_twograph>`.
Details are in ยง7E of [BvL84]_.
INPUT:
- ``q`` -- a power of an odd prime number
.. SEEALSO::
* :meth:`~sage.graphs.graph_generators.GraphGenerators.TaylorTwographDescendantSRG`
EXAMPLES::
sage: t=graphs.TaylorTwographSRG(3); t
Taylor two-graph SRG: Graph on 28 vertices
sage: t.is_strongly_regular(parameters=True)
(28, 15, 6, 10)
"""
G, l, v0 = TaylorTwographDescendantSRG(q, clique_partition=True)
G.add_vertex(v0)
G.seidel_switching(sum(l[:(q**2+1)/2],[]))
G.name("Taylor two-graph SRG")
return G
def sum(xs, y_from_x=lambda x: x):
'''
>>> range(10) | sum()
45
>>> range(10) | sum(lambda x: -x)
-45
'''
return __builtins__.sum(y_from_x(x) for x in xs)
def count(xs, predicate=lambda x: True):
'''
>>> [1, 2, 3] | count()
3
>>> range(10) | where(lambda x: x % 2 == 0) | count()
5
>>> [] | count()
0
'''
return __builtins__.sum(1 for x in xs | where(predicate))
def is_twograph(T):
r"""
Checks that the incidence system `T` is a two-graph
INPUT:
- ``T`` -- an :class:`incidence structure <sage.combinat.designs.IncidenceStructure>`
EXAMPLES:
a two-graph from a graph::
sage: from sage.combinat.designs.twographs import (is_twograph, TwoGraph)
sage: p=graphs.PetersenGraph().twograph()
sage: is_twograph(p)
True
a non-regular 2-uniform hypergraph which is a two-graph::
sage: is_twograph(TwoGraph([[1,2,3],[1,2,4]]))
True
TESTS:
wrong size of blocks::
sage: is_twograph(designs.projective_plane(3))
False
a triple system which is not a two-graph::
sage: is_twograph(designs.projective_plane(2))
False
"""
if not T.is_uniform(3):
return False
# A structure for a fast triple existence check
v_to_blocks = {v:set() for v in range(T.num_points())}
for B in T._blocks:
B = frozenset(B)
for x in B:
v_to_blocks[x].add(B)
def has_triple(x_y_z):
x, y, z = x_y_z
return bool(v_to_blocks[x] & v_to_blocks[y] & v_to_blocks[z])
# Check that every quadruple contains an even number of triples
from six.moves.builtins import sum
for quad in combinations(range(T.num_points()),4):
if sum(map(has_triple,combinations(quad,3))) % 2 == 1:
return False
return True
def is_twograph(T):
r"""
Checks that the incidence system `T` is a two-graph
INPUT:
- ``T`` -- an :class:`incidence structure <sage.combinat.designs.IncidenceStructure>`
EXAMPLES:
a two-graph from a graph::
sage: from sage.combinat.designs.twographs import (is_twograph, TwoGraph)
sage: p=graphs.PetersenGraph().twograph()
sage: is_twograph(p)
True
a non-regular 2-uniform hypergraph which is a two-graph::
sage: is_twograph(TwoGraph([[1,2,3],[1,2,4]]))
True
TESTS:
wrong size of blocks::
sage: is_twograph(designs.projective_plane(3))
False
a triple system which is not a two-graph::
sage: is_twograph(designs.projective_plane(2))
False
"""
if not T.is_uniform(3):
return False
# A structure for a fast triple existence check
v_to_blocks = {v:set() for v in range(T.num_points())}
for B in T._blocks:
B = frozenset(B)
for x in B:
v_to_blocks[x].add(B)
def has_triple(x_y_z):
x, y, z = x_y_z
return bool(v_to_blocks[x] & v_to_blocks[y] & v_to_blocks[z])
# Check that every quadruple contains an even number of triples
from six.moves.builtins import sum
for quad in combinations(range(T.num_points()),4):
if sum(map(has_triple,combinations(quad,3))) % 2 == 1:
return False
return True
def S(x, y):
return sum(x[j] * y[2 - j] ** q for j in range(3))
def P(x, y):
return sum(x[j] * y[m - 1 - j] ** q for j in range(m)) == 0