Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revisionBoth sides next revision | ||
tutorial:aut_of_graphs [2014/01/03 15:45] – external edit 127.0.0.1 | user_guide:aut_of_graphs [2019/01/25 09:27] – ↷ Links adapted because of a move operation oroehrig | ||
---|---|---|---|
Line 6: | Line 6: | ||
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: | 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); | + | polytope > $c=cube(2); |
- | $c-> | + | polytope > $c-> |
</ | </ | ||
- | {{:tutorial: | + | {{user_guide: |
- | To study the automorphisms of this graph, we create a '' | + | To study the automorphisms of this graph, we create a '' |
< | < | ||
- | $g=new props:: | + | polytope > $g=new props:: |
</ | </ | ||
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 '' | 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 '' | ||
Line 25: | Line 25: | ||
Now, we compute the generators of the automorphism group of this graph: | Now, we compute the generators of the automorphism group of this graph: | ||
< | < | ||
- | $aut=automorphisms($g); | + | polytope > $aut=automorphisms($g); |
</ | </ | ||
In this case, the automorphism group has two generators: | In this case, the automorphism group has two generators: | ||
Line 37: | Line 37: | ||
In order to be able to work with the group, we create a new Group object, which lives in the application '' | In order to be able to work with the group, we create a new Group object, which lives in the application '' | ||
< | < | ||
- | $autgroup=new group:: | + | polytope > $autgroup=new group:: |
</ | </ | ||
Now we can ask for basic properties of the group, e.g., the number of elements: | Now we can ask for basic properties of the group, e.g., the number of elements: | ||
Line 46: | Line 46: | ||
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: | 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< | + | polytope > $s0=new Set< |
- | $stab0=group:: | + | polytope > $stab0=group:: |
</ | </ | ||
We learn that the node 0 is only fixed by the permutation '' | We learn that the node 0 is only fixed by the permutation '' | ||
Line 58: | Line 58: | ||
In the second case, we look at the subgroup which leaves the set '' | In the second case, we look at the subgroup which leaves the set '' | ||
< | < | ||
- | $s12=new Set< | + | polytope > $s12=new Set< |
- | $stab12=group:: | + | polytope > $stab12=group:: |
</ | </ | ||
Now, we obtain a group of order 4: | Now, we obtain a group of order 4: |