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Solving a non-Cayley transitive graph embedding

Ed Pegg, Wolfram Research
Posted 1 day ago

{"NoncayleyTransitive",{28,13}} didn't have a known nice embedding. As a Local Graph, it was $C_9$, so I figured I could wrangle it.

And I was right! 

Graph[GraphData[{"NoncayleyTransitive",{28,13}}], VertexCoordinates->Join[Thread[{3,5,8,9,10,7,6,4,2}->(-1CirclePoints[9])],{1->{0,0}}, Thread[{18,13,15,12,11,16,14,17,19}->.7CirclePoints[9]],Thread[{26,25,21,22,23,20,24,27,28}->.5CirclePoints[9]]]]

enter image description here

I've found nice embeddings for many graphs, including things like the Hoffman-Singleton graph. In general, if a graph has a lot of symmetry, it has a nice embedding. It just needs to be found. I did not solve the 77-graph, but offering a $77 prize eventually led to a solution.

POSTED BY: Ed Pegg
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Perhaps better 

Graph[GraphData[{"NoncayleyTransitive",{28,13}}],VertexCoordinates->Join[Thread[{3,5,8,9,10,7,6,4,2}->(CirclePoints[9])],{1->{0,0}},Thread[{18,13,15,12,11,16,14,17,19}->Root[-1+3*#1+6*#1^2+#1^3&,2,0]CirclePoints[9]],Thread[{26,25,21,22,23,20,24,27,28}->Root[-1-6*#1-3*#1^2+#1^3&,2,0]CirclePoints[9]]]]

enter image description here
POSTED BY: Ed Pegg
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