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GraphTheory

  

AutomorphismGroup

  

compute the automorphism group

 

Calling Sequence

Parameters

Options

Definition of Automorphism Group

Details

Description

Examples

Compatibility

Calling Sequence

AutomorphismGroup( G, opts )

Parameters

G

-

a graph

opts

-

(optional) zero or more options as specified below

Options

• 

storage=rectangular, sparse, or auto

  

This option controls whether the dense or sparse algorithm from the Nauty library is used. The values rectangular and sparse correspond to the dense and sparse algorithms, respectively, while the value auto automatically chooses which algorithm to employ based on a heuristic depending on the number of vertices and edges in G. The default is auto.

Definition of Automorphism Group

• 

Let G be a graph with vertex set V.

• 

An automorphism σ of a graph G is a permutation of V such that for any pair of vertices u and v in V, there is a (directed) edge from u to v in G if and only if there is a (directed) edge from σu to σv.

• 

The set of automorphisms of G form a group. The group identity is the automorphism that is the identity mapping on V, and the group operation is function composition.

• 

No general polynomial-time algorithm for computing graph automorphisms is presently known.

Details

• 

This command makes use of the Nauty library for computing automorphism groups and canonical labelings.

Description

• 

The AutomorphismGroup( G ) command computes the group of automorphisms of a given graph G.

• 

The automorphism group is represented as a permutation group.

• 

The graph G may be directed or undirected, but must be unweighted.

Examples

withGraphTheory:withGroupTheory:

Compute the automorphism group of the cycle graph on 5 vertices and verify it is isomorphic to the dihedral group D5.

C5CycleGraph5

C5Graph 1: an undirected unweighted graph with 5 vertices and 5 edge(s)

(1)

GAutomorphismGroupC5

G1,23,5,2,53,4

(2)

AreIsomorphicG,DihedralGroup5

true

(3)

Compute the automorphism group of the complete graph on 4 vertices and verify it is isomorphic to the symmetric group S4.

K4CompleteGraph4

K4Graph 2: an undirected unweighted graph with 4 vertices and 6 edge(s)

(4)

GAutomorphismGroupK4

G1,2,2,3,3,4

(5)

AreIsomorphicG,SymmetricGroup4

true

(6)

Compute the automorphism group of the Petersen graph and display its order.

PGSpecialGraphs:-PetersenGraph

PGGraph 3: an undirected unweighted graph with 10 vertices and 15 edge(s)

(7)

GAutomorphismGroupPG

G1,23,56,97,8,2,53,47,108,9,3,94,87,10,4,75,68,10

(8)

GroupOrderG

120

(9)

Compatibility

• 

The GraphTheory[AutomorphismGroup] command was introduced in Maple 2017.

• 

For more information on Maple 2017 changes, see Updates in Maple 2017.

See Also

GraphTheory

GraphTheory[CanonicalGraph]

GroupTheory