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How to make colors of edges in CayleyGraph ?

Yoshihiro Sato
Posted 10 years ago

My question is how to make colors of edges in CayleyGraph. This is an example.

group = PermutationGroup[pp]
     PermutationGroup[{Cycles[{{5, 8, 6}}], Cycles[{{4, 7, 8, 6, 5}}], 
      Cycles[{{2, 5, 8, 6, 3}}], Cycles[{{1, 4, 7, 8, 6, 5, 2}}], 
      Cycles[{{1, 4, 7, 8, 6, 3, 2}}], Cycles[{{2, 5, 4, 7, 8, 6, 3}}], 
      Cycles[{{1, 4, 5, 8, 6, 3, 2}}]}]

cayleygraph = CayleyGraph[group];

There are seven colors of edges in CayleyGraph. My trial to make colors is as follows.

colors = Hue[#] & /@ ((1/7)*{0, 1, 2, 3, 4, 5, 6} // N)
   {Hue[0.], Hue[0.14285714285714285`], Hue[0.2857142857142857], 
    Hue[0.42857142857142855`], Hue[0.5714285714285714], Hue[0.7142857142857143], Hue[0.8571428571428571]}

I want to know whether it is correct or not for edge colors of CayleyGraph.

Thanks, Yoshihiro Sato
POSTED BY: Yoshihiro Sato
5 Replies
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Yoshihiro Sato
Yoshihiro Sato
Posted 10 years ago

Hello,
I could solve the problem using GroupElementToWord.

enter image description here

enter image description here

I would like to report a 15 puzzle solver, which uses Graph and PermutationGroup. Please access my blog in Japanese.
GridGraph[3,3]
GridGraph[4,4]
POSTED BY: Yoshihiro Sato 
CayleyGraph[     PermutationGroup[{Cycles[{{5, 8, 6}}], Cycles[{{4, 7, 8}}]}], 
 VertexLabels -> Placed["Name", Center], VertexSize -> 1.5]

enter image description here
POSTED BY: Simon Cadrin
Yoshihiro Sato
Yoshihiro Sato
Posted 10 years ago

Hello Simon,

Thank you for your suggestion.

Documentation of CayleyGraph says that this is only a useful representation for small groups, and for groups with a few hundred elements is generally already too complex.

I hope we can get a value of vertices and edges of CayleyGraph for large group if possible, although the graph is complicated.

I have tried to use a CayleyGraph for small group, which is very interesting for me.

Thanks, Yoshihiro Sato

enter image description here enter image description here

Attachments:
POSTED BY: Yoshihiro Sato
Yoshihiro Sato
Yoshihiro Sato
Posted 10 years ago

This group is the same as AlternatingGroup[8].

group = PermutationGroup[{Cycles[{{5, 8, 6}}], 
    Cycles[{{4, 7, 8, 6, 5}}], Cycles[{{2, 5, 8, 6, 3}}], 
    Cycles[{{1, 4, 7, 8, 6, 5, 2}}], Cycles[{{1, 4, 7, 8, 6, 3, 2}}], 
    Cycles[{{2, 5, 4, 7, 8, 6, 3}}], 
    Cycles[{{1, 4, 5, 8, 6, 3, 2}}]}];

group == AlternatingGroup[8]
 True
GroupOrder[group]
 20160
cayleygraph = CayleyGraph[group];
{VertexCount[cayleygraph], EdgeCount[cayleygraph]}
{20160, 141120}

The document of CayleyGraph says that generators are represented default using different colors, with increasing Hue values for the elements listed GroupGenerators. But it does not say how the Hue are calculated.

I have tries to use the following colors in CayleyGraph, but had no correct result. I wonder whether the color in CayleyGraph is correct. Anyone would tell me how to find out whether it is correct.

Thanks, Yoshihiro Sato

My trial: ShortestPath in CayleyGraph 1 -> 10 enter image description here

Attachments:
POSTED BY: Yoshihiro Sato 
CayleyGraph[PermutationGroup[{Cycles[{{1, 5, 4}}], Cycles[{{3, 4}}]}], VertexLabels -> Placed["Name", Center], VertexSize -> 0.8]

enter image description here
POSTED BY: Simon Cadrin
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