Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
sage: graphs.RandomGNP(20,0.5).antisymmetric()
False
sage: digraphs.RandomDirectedGNR(20,0.5).antisymmetric()
True
automorphism_group(partition=None, verbosity=0, edge_labels=False, order=False, return_group=True,
orbits=False)
Returns the largest subgroup of the automorphism group of the (di)graph whose orbit partition is finer than
the partition given. If no partition is given, the unit partition is used and the entire automorphism group is
given.
INPUT:
•partition - default is the unit partition, otherwise computes the subgroup of the full automorphism
group respecting the partition.
•edge_labels - default False, otherwise allows only permutations respecting edge labels.
•order - (default False) if True, compute the order of the automorphism group
•return_group - default True
•orbits - returns the orbits of the group acting on the vertices of the graph
Warning: Since trac ticket #14319 the domain of the automorphism group is equal to the graph’s
vertex set, and the translation argument has become useless.
OUTPUT: The order of the output is group, order, orbits. However, there are options to turn each of these
on or off.
EXAMPLES:
Graphs:
sage: graphs_query = GraphQuery(display_cols=[’graph6’],num_vertices=4)
sage: L = graphs_query.get_graphs_list()
sage: graphs_list.show_graphs(L)
sage: for g in L:
... G = g.automorphism_group()
... G.order(), G.gens()
(24, [(2,3), (1,2), (0,1)])
(4, [(2,3), (0,1)])
(2, [(1,2)])
(6, [(1,2), (0,1)])
(6, [(2,3), (1,2)])
(8, [(1,2), (0,1)(2,3)])
(2, [(0,1)(2,3)])
(2, [(1,2)])
(8, [(2,3), (0,1), (0,2)(1,3)])
(4, [(2,3), (0,1)])
(24, [(2,3), (1,2), (0,1)])
sage: C = graphs.CubeGraph(4)
sage: G = C.automorphism_group()
sage: M = G.character_table() # random order of rows, thus abs() below
sage: QQ(M.determinant()).abs()
712483534798848
sage: G.order()
384
1.1. Generic graphs 15