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trac #34319: merge with #34318
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@dcoudert
dcoudert committed Aug 20, 2022
2 parents 328cc6a + b62e29e commit dc4bcbd
Showing 1 changed file with 70 additions and 46 deletions.
116 changes: 70 additions & 46 deletions src/sage/graphs/generic_graph.py
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Original file line number Diff line number Diff line change
Expand Up @@ -21809,23 +21809,47 @@ def eigenvectors(self, laplacian=False):

sage: C = graphs.CycleGraph(8)
sage: C.eigenvectors()
[(2, [
(1, 1, 1, 1, 1, 1, 1, 1)
], 1), (-2, [
(1, -1, 1, -1, 1, -1, 1, -1)
], 1), (0, [
(1, 0, -1, 0, 1, 0, -1, 0),
(0, 1, 0, -1, 0, 1, 0, -1)
], 2), (-1.4142135623..., [(1, 0, -1, 1.4142135623..., -1, 0, 1, -1.4142135623...), (0, 1, -1.4142135623..., 1, 0, -1, 1.4142135623..., -1)], 2), (1.4142135623..., [(1, 0, -1, -1.4142135623..., -1, 0, 1, 1.4142135623...), (0, 1, 1.4142135623..., 1, 0, -1, -1.4142135623..., -1)], 2)]
[(2,
[
(1, 1, 1, 1, 1, 1, 1, 1)
],
1),
(-2,
[
(1, -1, 1, -1, 1, -1, 1, -1)
],
1),
(0,
[
(1, 0, -1, 0, 1, 0, -1, 0),
(0, 1, 0, -1, 0, 1, 0, -1)
],
2),
(-1.4142135623...,
[(1, 0, -1, 1.4142135623..., -1, 0, 1, -1.4142135623...),
(0, 1, -1.4142135623..., 1, 0, -1, 1.4142135623..., -1)],
2),
(1.4142135623...,
[(1, 0, -1, -1.4142135623..., -1, 0, 1, 1.4142135623...),
(0, 1, 1.4142135623..., 1, 0, -1, -1.4142135623..., -1)],
2)]

A digraph may have complex eigenvalues. Previously, the complex parts of
graph eigenvalues were being dropped. For a 3-cycle, we have::

sage: T = DiGraph({0:[1], 1:[2], 2:[0]})
sage: T.eigenvectors()
[(1, [
(1, 1, 1)
], 1), (-0.5000000000... - 0.8660254037...*I, [(1, -0.5000000000... - 0.8660254037...*I, -0.5000000000... + 0.8660254037...*I)], 1), (-0.5000000000... + 0.8660254037...*I, [(1, -0.5000000000... + 0.8660254037...*I, -0.5000000000... - 0.8660254037...*I)], 1)]
[(1,
[
(1, 1, 1)
],
1),
(-0.5000000000... - 0.8660254037...*I,
[(1, -0.5000000000... - 0.8660254037...*I, -0.5000000000... + 0.8660254037...*I)],
1),
(-0.5000000000... + 0.8660254037...*I,
[(1, -0.5000000000... + 0.8660254037...*I, -0.5000000000... - 0.8660254037...*I)],
1)]
"""
if laplacian:
M = self.kirchhoff_matrix(vertices=list(self))
Expand Down Expand Up @@ -21944,7 +21968,7 @@ def eigenspaces(self, laplacian=False):
# which would be a change in default behavior
return M.right_eigenspaces(format='galois', algebraic_multiplicity=False)

### Automorphism and isomorphism
# Automorphism and isomorphism

def relabel(self, perm=None, inplace=True, return_map=False, check_input=True, complete_partial_function=True, immutable=None):
r"""
Expand Down Expand Up @@ -22170,14 +22194,14 @@ def relabel(self, perm=None, inplace=True, return_map=False, check_input=True, c
if not inplace:
G = copy(self)
perm2 = G.relabel(perm,
return_map= return_map,
check_input = check_input,
complete_partial_function = complete_partial_function)
return_map=return_map,
check_input=check_input,
complete_partial_function=complete_partial_function)

if immutable is None:
immutable = self.is_immutable()
if immutable:
G = self.__class__(G, immutable = True)
G = self.__class__(G, immutable=True)

if return_map:
return G, perm2
Expand Down Expand Up @@ -22290,8 +22314,8 @@ def degree_to_cell(self, vertex, cell):
(0, 2)
"""
if self._directed:
in_neighbors_in_cell = set([a for a,_,_ in self.incoming_edges(vertex)]) & set(cell)
out_neighbors_in_cell = set([a for _,a,_ in self.outgoing_edges(vertex)]) & set(cell)
in_neighbors_in_cell = set([a for a, _, _ in self.incoming_edges(vertex)]) & set(cell)
out_neighbors_in_cell = set([a for _, a, _ in self.outgoing_edges(vertex)]) & set(cell)
return (len(in_neighbors_in_cell), len(out_neighbors_in_cell))
else:
neighbors_in_cell = set(self.neighbors(vertex)) & set(cell)
Expand Down Expand Up @@ -22351,9 +22375,9 @@ def is_equitable(self, partition, quotient_matrix=False):
"""
from sage.misc.flatten import flatten
if sorted(flatten(partition, max_level=1)) != self.vertices(sort=True):
raise TypeError("Partition (%s) is not valid for this graph: vertices are incorrect."%partition)
if any(len(cell)==0 for cell in partition):
raise TypeError("Partition (%s) is not valid for this graph: there is a cell of length 0."%partition)
raise TypeError("Partition (%s) is not valid for this graph: vertices are incorrect." % partition)
if any(not cell for cell in partition):
raise TypeError("Partition (%s) is not valid for this graph: there is a cell of length 0." % partition)
if quotient_matrix:
from sage.matrix.constructor import Matrix
from sage.rings.integer_ring import IntegerRing
Expand Down Expand Up @@ -22439,9 +22463,9 @@ def coarsest_equitable_refinement(self, partition, sparse=True):
"""
from sage.misc.flatten import flatten
if set(flatten(partition, max_level=1)) != set(self):
raise TypeError("partition (%s) is not valid for this graph: vertices are incorrect"%partition)
raise TypeError("partition (%s) is not valid for this graph: vertices are incorrect" % partition)
if any(len(cell) == 0 for cell in partition):
raise TypeError("partition (%s) is not valid for this graph: there is a cell of length 0"%partition)
raise TypeError("partition (%s) is not valid for this graph: there is a cell of length 0" % partition)
if self.has_multiple_edges():
raise TypeError("refinement function does not support multiple edges")
G = copy(self)
Expand Down Expand Up @@ -22697,8 +22721,8 @@ def automorphism_group(self, partition=None, verbosity=0,
raise NotImplementedError("algorithm 'bliss' cannot be used for graph with multiedges")
have_bliss = False

if (algorithm == 'bliss' or # explicit choice from the user; or
(algorithm is None and # by default
if (algorithm == 'bliss' or # explicit choice from the user; or
(algorithm is None and # by default
have_bliss)):

Bliss().require()
Expand All @@ -22723,8 +22747,7 @@ def automorphism_group(self, partition=None, verbosity=0,
return ret[0]
return ret

if (algorithm is not None and
algorithm != "sage"):
if algorithm is not None and algorithm != "sage":
raise ValueError("'algorithm' must be equal to 'bliss', 'sage', or None")

from sage.groups.perm_gps.partn_ref.refinement_graphs import search_tree
Expand All @@ -22738,21 +22761,24 @@ def automorphism_group(self, partition=None, verbosity=0,
partition = [list(self)]

if edge_labels or self.has_multiple_edges():
G, partition, relabeling = graph_isom_equivalent_non_edge_labeled_graph(self, partition, return_relabeling=True, ignore_edge_labels=(not edge_labels))
ret = graph_isom_equivalent_non_edge_labeled_graph(self, partition,
return_relabeling=True,
ignore_edge_labels=(not edge_labels))
G, partition, relabeling = ret
G_vertices = list(chain(*partition))
G_to = {u: i for i,u in enumerate(G_vertices)}
G_to = {u: i for i, u in enumerate(G_vertices)}
DoDG = DiGraph if self._directed else Graph
H = DoDG(len(G_vertices), loops=G.allows_loops())
HB = H._backend
for u,v in G.edge_iterator(labels=False):
for u, v in G.edge_iterator(labels=False):
HB.add_edge(G_to[u], G_to[v], None, G._directed)
GC = HB.c_graph()[0]
partition = [[G_to[vv] for vv in cell] for cell in partition]
A = search_tree(GC, partition, lab=False, dict_rep=True, dig=dig, verbosity=verbosity, order=order)
if order:
a,b,c = A
a, b, c = A
else:
a,b = A
a, b = A
b_new = {v: b[G_to[v]] for v in G_to}
b = b_new
# b is a translation of the labellings
Expand Down Expand Up @@ -22781,27 +22807,27 @@ def automorphism_group(self, partition=None, verbosity=0,
b = translation_d
else:
G_vertices = list(chain(*partition))
G_to = {u: i for i,u in enumerate(G_vertices)}
G_to = {u: i for i, u in enumerate(G_vertices)}
DoDG = DiGraph if self._directed else Graph
H = DoDG(len(G_vertices), loops=self.allows_loops())
HB = H._backend
for u,v in self.edge_iterator(labels=False):
for u, v in self.edge_iterator(labels=False):
HB.add_edge(G_to[u], G_to[v], None, self._directed)
GC = HB.c_graph()[0]
partition = [[G_to[vv] for vv in cell] for cell in partition]

if return_group:
A = search_tree(GC, partition, dict_rep=True, lab=False, dig=dig, verbosity=verbosity, order=order)
if order:
a,b,c = A
a, b, c = A
else:
a,b = A
a, b = A
b_new = {v: b[G_to[v]] for v in G_to}
b = b_new
else:
a = search_tree(GC, partition, dict_rep=False, lab=False, dig=dig, verbosity=verbosity, order=order)
if order:
a,c = a
a, c = a

output = []
if return_group:
Expand All @@ -22825,17 +22851,15 @@ def automorphism_group(self, partition=None, verbosity=0,
from sage.groups.perm_gps.partn_ref.refinement_graphs import get_orbits
output.append([[G_from[v] for v in W] for W in get_orbits(a, self.num_verts())])

# A Python switch statement!
return { 0: None,
1: output[0],
2: tuple(output),
3: tuple(output),
4: tuple(output)
}[len(output)]
if len(output) == 1:
return output[0]
elif len(output) > 1:
return tuple(output)
return None

def is_vertex_transitive(self, partition=None, verbosity=0,
edge_labels=False, order=False,
return_group=True, orbits=False):
edge_labels=False, order=False,
return_group=True, orbits=False):
"""
Returns whether the automorphism group of self is transitive within
the partition provided, by default the unit partition of the
Expand Down

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