The automorphism group functionality is an interface to version 2.6r7 of the nauty and {Traces} packages of B. McKay and A. Piperno. For a paper describing nauty see [McK81]. There also exists a user's manual [MP] describing the essential features of nauty and Traces.
Compute the automorphism group of the graph G. This returns the group A, the vertex and edge G-sets (in that order), the power structure (A) of all automorphisms of G, and the transfer map from A to (A). Note that G may be directed or undirected. For small graphs (i.e. having less than 500 vertices) the canonically labelled graph is also returned.The automorphism group computation may be driven by several user parameters. There are five principal parameters: Al, Canonical, Stabilizer, Invariant and Print. Some of these parameters have associated parameters.
Al: MonStgElt Default: "nauty"The parameter Al controls whether the nauty or Traces algorithms are used to compute the automorphism group. Some parameters are only applicable with the nauty algorithm.Canonical: BoolElt Default: {false}If the parameter Canonical is set to true, then the canonically labelled graph will be returned. If the graph has fewer than 500 vertices then the default value for this parameter is true, while if the graph has 500 or more vertices, the default value is false. It is important to note that the canonically labelled graph is dependent upon all the parameters used to compute the automorphism group, including the algorithm parameter.Stabilizer: [ { GrphVert } ] Default:If the parameter Stabilizer is assigned a partition P of the vertex-set of G as its value, then the subgroup of the automorphism group of G that preserves the partition P will be computed. The partition given as Stabilizer need not be a partition of the full vertex-set of G. Any vertices not covered by the sets in Stabilizer are assumed to belong to an extra partition cell.Invariant: MonStgElt Default:This parameter is only relevant when the nauty algorithm is used. It causes the algorithm to use the named invariant to assist the auto group computation. The invariant is specified by a string which is the name of a C function computing an invariant, as on pages 37--39 of the nauty and Traces manual [MP]. The default value for Invariant is "Null" when G is an undirected graph, or "adjacencies" when G is a digraph. For a more detailed discussion on invariants see Subsection nauty Invariants.There are three optional associated parameters to Invariant:
Minlevel: RngIntElt Default: 0Maxlevel: RngIntElt Default:Arg: RngIntElt Default:An expression, depending upon the type of invariant. When G is undirected these three parameters take the values 0, 1, and 0 respectively. When G is a digraph they take the values 0, 2, and 0 respectively. When Invariant is one of "indsets", "cliques", "cellcliq", or "cellind", the default value for Arg is 3. Again, see Subsection nauty Invariants for more information.Print: RngIntElt Default: 0The integer valued parameter Print is used to control informative printing according to the following table:Print := 0 : No output (default).
Print := 1 : Statistics are printed.
Print := 2 : Summary upon completion of each level (nauty only).
Print := 3 : Print the automorphisms as they are discovered.
IgnoreLabels: BoolElt Default: falseBy default, the automorphism group computation respects vertex labels (colours). That is, elements of the group will only take a vertex to an identically labelled vertex. The Boolean value IgnoreLabels will treat all vertices as identically labelled for the purposes of this computation.
Invariants can be used to supplement the built-in partition refinement code in nauty, to assist in the processing of difficult graphs. Invariants are used with three parameters:
Minlevel: the minimum depth in the search tree at which an invariant is to be applied (default 0)
Maxlevel: the maximum depth in the search tree at which an invariant is to be applied (default 1 for undirected graphs and 2 for digraphs)
Arg: an integer argument which has a different meaning (or no meaning) for each invariant
The root of the tree, corresponding to the partition provided by the user, has level 1. Note that the invariants use an existing partition and attempt to refine it. Thus it can be that an invariant fails to refine the partition at the root of the search tree but succeeds further down the tree. In particular, all invariants necessarily fail at the root of the tree if the graph is vertex-transitive.
The invariants are :
Returns true if and only the invariant in Invariant could refine the graph G's vertex-set partition. If no invariant is set in Invariant then the refinement procedure which is tested is taken to be the default one.Four parameters can be passed:
Stabilizer: [ { Vert } ] Default:Invariant: MonStgElt Default:Arg: RngIntElt Default:IgnoreLabels: BoolElt Default: falseThe parameters have the same meaning as for AutomorphismGroup.
For any calls to nauty or Traces via Magma functions the graph colouring of a graph G (as set by AssignLabels or a similar function) is taken as an intrinsic property of G. Consequently, the default automorphism group computed by nauty or Traces is the group for the coloured graph G. In particular this implies that, when G is coloured, this default automorphism group is not G's full automorphism group.
There are several Magma functions which make use of the automorphism group of a graph, and therefore rely on a call to nauty. We list them here:
The functions in Group 2 all use the graph's default group. If a group A of the graph G has already been computed but is not G's default group (for example if a stabilizer of G was given when computing A) then the graph's default group will be computed. Consequently, as an example, if G is coloured, IsTransitive(G) will (always) return false.
For all these functions it is possible to drive the group computation by setting an appropriate Invariant parameter. This is particularly useful when it is known that some invariant will speed up the group computation. This is achieved by first computing the group using AutomorphismGroup while setting Invariant. This parameter is then remembered for any further (re)computation of the default automorphism group, unless reset by a subsequent call to AutomorphismGroup. For convenience, the function IsIsomorphic accepts a parameter suite very similar to that of the AutomorphismGroup function: It is particularly useful when stabilizers need to be set.
Given a graph G, return the canonically labelled graph isomorphic to G. To see which automorphism group is computed see Subsection Graph Colouring and Automorphism Group. Users should be aware that the canonical graph is dependent upon the invariant used when computing the automorphism group. Thus, the computation of the automorphism group of G using different invariants will produce different canonical labellings.
Returns the automorphism group of the graph G in its action on the edges of G, and the G-set of edges of G. To see which automorphism group is computed see Subsection Graph Colouring and Automorphism Group.
This function returns true if the graphs G and H are isomorphic. If G and H are isomorphic, a second value is returned, namely a mapping between the vertices of G and the vertices of H giving the isomorphism.The parameters accepted by IsIsomorphic are almost the same as for
AutomorphismGroup. The exceptions are:Stabilizer: [ { Vert } ] Default:This sets the stabilizer set for both graphs G and H. If one wishes to set two distinct stabilizers then one would also use:Stabilizer2: [ { Vert } ] Default:This sets the stabilizer set for the second graph H only.IgnoreLabels: BoolElt Default: falseThis indicates whether or not to ignore the graph labelling for both graphs G and H.Graph isomorphism is understood as mapping colours to colours and stabilizers to stabilizers. Consequently comparing two graphs for isomorphism will always return false if one graph is coloured while the other one is not. Similarly two graphs can't be isomorphic if each is coloured using a different set of colours. Moreover, it is a nauty requirement that in order to achieve mapping of stabilizers they must be compatible: that is, they must have same-sized cells in the same order.
In general, to see which and how the graphs' automorphism groups are computed see Subsection Graph Colouring and Automorphism Group.
> G := Graph<5 | { {1,2}, {2,3}, {3,4}, {4,5}, {5,1} }>;
>
> AssignLabels(VertexSet(G), ["a", "b", "a", "b", "b"]);
> A, _, _, _, _, C1 := AutomorphismGroup(G);
> A;
Permutation group A acting on a set of cardinality 5
Order = 2
(1, 3)(4, 5)
> C1;
Graph
Vertex Neighbours
1 3 5 ;
2 4 5 ;
3 1 4 ;
4 2 3 ;
5 1 2 ;
> B, _, _, _, _, C2 := AutomorphismGroup(G : IgnoreLabels := true);
> B;
Permutation group B acting on a set of cardinality 5
Order = 10 = 2 * 5
(2, 5)(3, 4)
(1, 2)(3, 5)
> C2;
Graph
Vertex Neighbours
1 2 3 ;
2 1 4 ;
3 1 5 ;
4 2 5 ;
5 3 4 ;
> C, _, _, _, _, C3 := AutomorphismGroup(G : Stabilizer :=
> [ { VertexSet(G) | 1, 2 } ]);
> C;
Permutation group C acting on a set of cardinality 5
> #C;
1
> C3;
Graph
Vertex Neighbours
1 3 4 ;
2 3 5 ;
3 1 2 ;
4 1 5 ;
5 2 4 ;
We now check that the graph returned by CanonicalGraph
is identical to C1:
> C4 := CanonicalGraph(G); > C4 eq C1; true
> G1 := CompleteGraph(5);
> AssignLabels(VertexSet(G1), ["a", "a", "b", "b", "b"]);
> G2 := CompleteGraph(5);
> IsIsomorphic(G1, G2);
false
> IsIsomorphic(G1, G2 : IgnoreLabels := true);
true
>
> V1 := Vertices(G1);
> V2 := Vertices(G2);
> S1 := { V1 | 1, 2, 3};
> S2 := { V2 | 3, 4, 5};
>
> IsIsomorphic(G1, G2 : Stabilizer := [S1],
> Stabilizer2 := [S2]);
false
> IsIsomorphic(G1, G2 : Stabilizer := [S1],
> Stabilizer2 := [S2], IgnoreLabels := true);
true
> IsIsomorphic(G1, G2 : Stabilizer := [S1],
> IgnoreLabels := true);
true
>
> SS1 := [ { V1 | 1}, {V1 | 2, 3} ];
> SS2 := [ { V2 | 3, 4}, { V2 | 1} ];
> IsIsomorphic(G1, G2 : Stabilizer := SS1, Stabilizer2 := SS2,
> IgnoreLabels := true);
false
>
> SS1 := [ {V1 | 2, 3}, { V1 | 1} ];
> IsIsomorphic(G1, G2 : Stabilizer := SS1, Stabilizer2 := SS2,
> IgnoreLabels := true);
true
The second example shows that two graphs are isomorphic if and only
if colours map to the same colours.
> G1 := CompleteGraph(5); > AssignLabels(VertexSet(G1), ["b", "b", "b", "a", "a"]); > G2 := CompleteGraph(5); > AssignLabels(VertexSet(G2), ["a", "a", "b", "b", "b"]); > IsIsomorphic(G1, G2); true > > G1 := CompleteGraph(5); > AssignLabels(VertexSet(G1), ["a", "b", "b", "c", "c"]); > G2 := CompleteGraph(5); > AssignLabels(VertexSet(G2), ["a", "c", "b", "b", "c"]); > IsIsomorphic(G1, G2); true > > G1 := CompleteGraph(5); > G2 := CompleteGraph(5); > AssignLabels(VertexSet(G1), ["a", "a", "b", "b", "b"]); > AssignLabels(VertexSet(G2), ["b", "b", "a", "a", "a"]); > IsIsomorphic(G1, G2); false > IsIsomorphic(G1, G2 : IgnoreLabels := true); true > > G1 := CompleteGraph(5); > AssignLabels(VertexSet(G1), ["b", "b", "b", "a", "a"]); > G2 := CompleteGraph(5); > AssignLabels(VertexSet(G2), ["b", "b", "c", "c", "c"]); > IsIsomorphic(G1, G2); false > IsIsomorphic(G1, G2 : IgnoreLabels := true); true
The automorphism group A of a graph G is given in its action on the standard support and it does not act directly on G. The action of A on G is obtained using the G-set mechanism. The two basic G-sets associated with the graph correspond to the action of A on the set of vertices V and the set of edges V of G. These two G-sets are given as return values of the function AutomorphismGroup or may be constructed directly. Additional G-sets associated with a graph may be built using the G-set constructors. Given a G-set Y for A, the action of A on Y may be studied using the permutation group functions that allow a G-set as an argument In this section, only a few of the available functions are described: see the section on G-sets for a complete list.
Let a be an element of the automorphism group A for the graph G and let Y be a G-set for A. Given an element y belonging either to Y or to a G-set derived from Y, find the image of y under a.
Let A be a subgroup of the automorphism group for the graph G and let Y be a G-set for A. Given an element y belonging either to Y or to a G-set derived from Y, construct the orbit of y under A.
Let A be a subgroup of the automorphism group for the graph G and let Y be a G-set for G. This function constructs the orbits of the action of A on Y.
Let A be a subgroup of the automorphism group for the graph G and let Y be a G-set for A. Given an element y belonging either to Y or to a G-set derived from Y, construct the stabilizer of y in A.
Given a subgroup A of the automorphism group of the graph G, and a G-set Y for A, construct the homomorphism φ: A -> L, where the permutation group L gives the action of A on the set Y. The function returns:
- (a)
- The natural homomorphism φ: A -> L;
- (b)
- The induced group L;
- (c)
- The kernel of the action (a subgroup of A).
Given a subgroup A of the automorphism group of the graph structure G, and a G-set Y for A, construct the permutation group L giving the action of A on the set Y.
Given a subgroup A of the automorphism group of the graph G, and a G-set Y for A, construct the kernel of the action of A on the set Y.
> S := { 1 .. 5 };
> V := &join[ Subsets({1..5}, 2*k) : k in [0..#S div 2]];
> E := { {u,v} : u,v in V | #(u sdiff v) eq 4 };
> G, V, E := StandardGraph( Graph< V | E >);
> G;
Graph
Vertex Neighbours
1 4 6 8 10 12 ;
2 5 6 9 12 14 ;
3 9 10 12 13 16 ;
4 1 7 9 14 16 ;
5 2 7 8 10 16 ;
6 1 2 13 15 16 ;
7 4 5 12 13 15 ;
8 1 5 9 11 13 ;
9 2 3 4 8 15 ;
10 1 3 5 14 15 ;
11 8 12 14 15 16 ;
12 1 2 3 7 11 ;
13 3 6 7 8 14 ;
14 2 4 10 11 13 ;
15 6 7 9 10 11 ;
16 3 4 5 6 11 ;
> A, AV, AE := AutomorphismGroup(G);
> A;
Permutation group aut acting on a set of cardinality 16
Order = 1920 = 2^7 * 3 * 5
(2, 15)(5, 11)(7, 14)(10, 12)
(3, 11)(8, 10)(9, 14)(13, 15)
(2, 11)(5, 15)(6, 8)(9, 16)
(2, 7)(4, 6)(9, 13)(14, 15)
(1, 2, 13, 15)(3, 11, 4, 5)(7, 10, 12, 14)(8, 9)
> CompositionFactors(A);
G
| Cyclic(2)
*
| Alternating(5)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
1
> IsPrimitive(A);
true
From the composition factors we guess that the group is Sym(5)
extended by the elementary abelian group of order 16. Let us
verify this and relate it to the graph. We begin by trying to get
at the group of order 16. We ask for the Fitting subgroup.
> F := FittingSubgroup(A);
> F;
Permutation group F acting on a set of cardinality 16
Order = 16 = 2^4
(1, 13)(2, 11)(3, 4)(5, 15)(6, 8)(7, 10)(9, 16)(12, 14)
(1, 10)(2, 9)(3, 12)(4, 14)(5, 8)(6, 15)(7, 13)(11, 16)
(1, 11)(2, 13)(3, 5)(4, 15)(6, 14)(7, 9)(8, 12)(10, 16)
(1, 12)(2, 6)(3, 10)(4, 7)(5, 16)(8, 11)(9, 15)(13, 14)
Since A is primitive, this looks as if it is an elementary
abelian regular normal subgroup:
> EARNS(A) eq F; trueThus, A has a regular normal subgroup N of order 16 which acts transitively on the vertices. Therefore, the stabilizer of any point is a complement of N.
> S1 := Stabilizer(A, AV, V!1);
> #S1;
120
> IsTransitive(S1);
false
> Orbits(S1);
[
GSet{@ 1 @},
GSet{@ 3, 7, 9, 11, 15 @},
GSet{@ 2, 5, 4, 14, 16, 6, 13, 12, 8, 10 @}
]
> exists(orbit){x : x in Orbits(S1) | #x eq 5};
true
> orbit;
GSet{@ 3, 7, 9, 11, 15 @}
> IsSymmetric(ActionImage(S1, orbit));
true
So the stabilizer of the vertex 1 is Sym(5), acting
faithfully on the 5 neighbours of vertex 1.