user_guide:tutorials:aut_of_graphs

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tutorial:aut_of_graphs [2014/01/03 15:45] – external edit 127.0.0.1user_guide:tutorials:aut_of_graphs [2019/01/28 17:40] – ↷ Links adapted because of a move operation oroehrig
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 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:
 <code> <code>
-$c=cube(2); +polytope > $c=cube(2); 
-$c->GRAPH->VISUAL;+polytope > $c->GRAPH->VISUAL;
 </code> </code>
-{{:tutorial:square.png?200|}}+{{user_guide:square.png?200|}}
  
-To study the automorphisms of this graph, we create a ''props::Graph'' object refering to the ''C++'' class named ''Graph'' (see the [[tutorial::apps_graph|tutorial on graphs]] for more details):+To study the automorphisms of this graph, we create a ''props::Graph'' object refering to the ''C++'' class named ''Graph'' (see the [[user_guide:tutorials:apps_graph|tutorial on graphs]] for more details):
 <code> <code>
-$g=new props::Graph($c->GRAPH->ADJACENCY);+polytope > $g=new props::Graph($c->GRAPH->ADJACENCY);
 </code> </code>
 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 ''$c->GRAPH->ADJACENCY'' or just the props::Graph object ''$g'': 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 ''$c->GRAPH->ADJACENCY'' or just the props::Graph object ''$g'':
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 Now, we compute the generators of the automorphism group of this graph: Now, we compute the generators of the automorphism group of this graph:
 <code> <code>
-$aut=automorphisms($g);+polytope > $aut=automorphisms($g);
 </code> </code>
 In this case, the automorphism group has two generators: In this case, the automorphism group has two generators:
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 In order to be able to work with the group, we create a new Group object, which lives in the application ''group'': In order to be able to work with the group, we create a new Group object, which lives in the application ''group'':
 <code> <code>
-$autgroup=new group::Group(GENERATORS=>$aut);+polytope > $autgroup=new group::Group(GENERATORS=>$aut);
 </code> </code>
 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:
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 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:
 <code> <code>
-$s0=new Set<Int>(0); +polytope > $s0=new Set<Int>(0); 
-$stab0=group::stabilizer_of_set($autgroup,$s0);+polytope > $stab0=group::stabilizer_of_set($autgroup,$s0);
 </code> </code>
 We learn that the node 0 is only fixed by the permutation ''0 2 1 3'': We learn that the node 0 is only fixed by the permutation ''0 2 1 3'':
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 In the second case, we look at the subgroup which leaves the set ''{1,2}'' invariant:  In the second case, we look at the subgroup which leaves the set ''{1,2}'' invariant: 
 <code> <code>
-$s12=new Set<Int>(1,2); +polytope > $s12=new Set<Int>(1,2); 
-$stab12=group::stabilizer_of_set($autgroup,$s12);+polytope > $stab12=group::stabilizer_of_set($autgroup,$s12);
 </code> </code>
 Now, we obtain a group of order 4: Now, we obtain a group of order 4:
  • user_guide/tutorials/aut_of_graphs.txt
  • Last modified: 2019/02/11 23:09
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