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GraphTheory

  

AutomorphismGroup

  

compute the automorphism group

 

Calling Sequence

Parameters

Options

Description

Definition of Automorphism Group

Details

Examples

Compatibility

Calling Sequence

AutomorphismGroup( G, opts )

Parameters

G

-

a graph

opts

-

(optional) zero or more options as specified below

Options

• 

partition=set of sets

  

This option specifies a partition P of the vertices of G and restricts the automorphism group returned to those automorphisms which preserve this partition; that is, for each vertex set P[i] in P and each vertex v in P[i], only those automorphisms which map v to a vertex in P[i] are included.

  

The partition is permitted to be a proper subset of the vertices of G; in this case, any vertex not included in the partition form an additional implicitly defined subset.

  

The default is the empty set, meaning that no restrictions are imposed on the automorphisms returned.

• 

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 means that Maple automatically determines which algorithm to employ based on a heuristic depending on the number of vertices and edges in G. The default is auto.

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.

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.

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

G3,4,1,2,2,3

(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

G2,53,47,108,9,3,94,87,10,4,75,68,10,1,23,56,97,8

(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.

• 

The GraphTheory[AutomorphismGroup] command was updated in Maple 2020.

• 

The partition option was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

GraphTheory

GraphTheory[CanonicalGraph]

GroupTheory