Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
sage: C.automorphism_group(orbits=True, return_group=False)
[[’000’, ’001’, ’010’, ’011’, ’100’, ’101’, ’110’, ’111’]]
TESTS:
We get a KeyError when given an invalid partition (trac #6087):
sage: g=graphs.CubeGraph(3)
sage: g.relabel()
sage: g.automorphism_group(partition=[[0,1,2],[3,4,5]])
Traceback (most recent call last):
...
KeyError: 6
Labeled automorphism group:
sage: digraphs.DeBruijn(3,2).automorphism_group()
Permutation Group with generators [(’01’,’02’)(’10’,’20’)(’11’,’22’)(’12’,’21’), (’00’,’11’)
sage: d = digraphs.DeBruijn(3,2)
sage: d.allow_multiple_edges(True)
sage: d.add_edge(d.edges()[0])
sage: d.automorphism_group()
Permutation Group with generators [(’01’,’02’)(’10’,’20’)(’11’,’22’)(’12’,’21’)]
The labeling is correct:
sage: g = graphs.PetersenGraph()
sage: ag = g.automorphism_group()
sage: for u,v in g.edges(labels = False):
... if len(ag.orbit((u,v),action="OnPairs")) != 30:
... print "ARggggggggggggg !!!"
Empty group, correct domain:
sage: Graph({’a’:[’a’], ’b’:[]}).automorphism_group()
Permutation Group with generators [()]
sage: Graph({’a’:[’a’], ’b’:[]}).automorphism_group().domain()
{’a’, ’b’}
average_degree()
Returns the average degree of the graph.
The average degree of a graph G = (V, E) is equal to \frac {2|E|}{|V|}.
EXAMPLES:
The average degree of a regular graph is equal to the degree of any vertex:
sage: g = graphs.CompleteGraph(5)
sage: g.average_degree() == 4
True
The average degree of a tree is always strictly less than 2:
sage: g = graphs.RandomGNP(20,.5)
sage: tree = Graph()
sage: tree.add_edges(g.min_spanning_tree())
sage: tree.average_degree() < 2
True
For any graph, it is equal to \frac {2|E|}{|V|}:
1.1. Generic graphs 17