This tutorial is probably also available as a Jupyter notebook in the `demo`

folder in the polymake source and on github.

Different versions of this tutorial: latest release, release 4.11, release 4.10, release 4.9, release 4.8, release 4.7, release 4.6, release 4.5, release 4.4, release 4.3, release 4.2, release 4.1, release 4.0, release 3.6, nightly master

# Automorphisms of Graphs

## First Example: The graph of a square

Let's look at the graph of a square. Since a square is a 2-cube, we can create the polytope and look at its graph:

> $c=cube(2); > $c->GRAPH->VISUAL;

**Transparency**

**Rotation**

**Display**

**Camera**

**SVG**

To study the automorphisms of this graph, we create a `GraphAdjacency<Dir>`

object refering to the `C++`

class named `Graph`

(see the tutorial on graphs for more details):

> $g=new GraphAdjacency($c->GRAPH->ADJACENCY);

The picture of the graph shows that the node with label 0 is adjacent to the nodes 1 and 2, Node 1 is adjacent to 0 and 3, and so on. For the complete adjacency information you can print `$c->GRAPH->ADJACENCY`

or just the GraphAdjacency object `$g`

:

> print rows_labeled($g); 0:1 2 1:0 3 2:0 3 3:1 2

Now, we compute the generators of the automorphism group of this graph (see the tutorial on groups for more info):

> $aut=automorphisms($g);

In this case, the automorphism group has two generators:

> print $aut; 0 2 1 3 1 0 3 2

Each generator is a permutation on the nodes. The first generator fixes the nodes 0 and 3, and exchanges the nodes 1 and 2, i.e., it describes the reflection along the diagonal through 0 and 3. The second generator is the reflection along the horizontal line.

In order to be able to work with the group, we create a new Group object, which lives in the application `group`

:

> $action = new group::PermutationAction(GENERATORS => $aut); > $autgroup = new group::Group(PERMUTATION_ACTION => $action);

Now we can ask for basic properties of the group, e.g., the number of elements:

> print $autgroup->ORDER; 8

Sometimes, it is useful to know which elements of the group fix a specific set of indices, that is, we are interested in the subgroup which is the stabilizer of the given set. In the first case, we just fix the index 0:

> $s0=new Set<Int>(0); > $stab0=group::stabilizer_of_set($action,$s0);

We learn that the node 0 is only fixed by the permutation `0 2 1 3`

:

> print $stab0->ORDER; 2 > print $stab0->PERMUTATION_ACTION->GENERATORS; 0 2 1 3

In the second case, we look at the subgroup which leaves the set `{1,2}`

invariant:

> $s12=new Set<Int>(1,2); > $stab12=group::stabilizer_of_set($action,$s12);

Now, we obtain a group of order 4:

> print $stab12->ORDER; 4 > print $stab12->PERMUTATION_ACTION->GENERATORS; 3 1 2 0 0 2 1 3

Finally, we compute the orbits of the indices under the three different groups:

> print $stab0->PERMUTATION_ACTION->ORBITS; {0} {1 2} {3} > print $stab12->PERMUTATION_ACTION->ORBITS; {0 3} {1 2} > print $autgroup->PERMUTATION_ACTION->ORBITS; {0 1 2 3}