Polyhedra from Cayley Graphs

Question

I was playing around with the Cayley graphs for some simple groups today and stumbled across something interesting, but can't quite figure out if there's something deeper going on. Here's what I did:

Consider the multiplicative group of the integers mod p, $\mathbb{Z}_{p}^{\times}$. We can generate $\mathbb{Z}_{p}^{\times}$ with any single element and obtain a simple Cayley graph. However, consider the Cayley graph generated by all primes strictly less than p, i.e. let $S=\{[a]: a \text{ is prime and } a<p\}$ and let $\Gamma_{p}$ be the Cayley graph $\Gamma_{p} = (\mathbb{Z}_{p}^{\times}, S)$.

Here is the Mathematica code I was using to generate some of the graphs:

plotGraph[p_] := (
  primesN := Table[Prime[n], {n, PrimePi[p] - 1}]; (*get generators*)

  (*function to compute adjacency matrix entries*)
  f[i_, j_] := (If[MemberQ[Mod[i*primesN, p], j], Return[1]];0); 

  M := Array[f, {p - 1, p - 1}]; (*create adjacency matrix*)
  MatrixForm[M] 
  GraphPlot3D[M, VertexLabeling -> True]
  )

I noticed that if n is not prime (and of course, $\mathbb{Z}_{n}-\{[0]\}$ is not a group), then the graph $\Gamma_{n}$ is really not interesting. However, when p is prime, the graphs have some nice structure. p=2 is a point, p=3 is a line segment, p=5 a square, p=7 an octahedron, p=11 looks like a pentagonal antiprism. However, I don't know if there is a pattern, or basically what's going on here. Does anyone have any insight?

Answer 1

This does not exactly answer but... there's pictures! :)

I changed to code into

Needs["TetGenLink`"];
plot[p_] := Module[{pts},
  pts = First@ Cases[
     GraphPlot3D@ Flatten@
       Table[i -> Mod[i q, p], 
         {i, 1, p - 1}, {q, Prime[Range[PrimePi[p] - 1]]}],
     Pattern[s, Rule[VertexCoordinateRules, c_]] :> c, Infinity];
  Graphics3D@ GraphicsComplex[pts, Polygon[TetGenConvexHull[pts]]]
  ]

Now saying

GraphicsGrid[Partition[plot /@ Prime[Range[3, 11]], 3]]

gets us

TetGenConvexHull only gets us a triangulation of the convex hull, so there are edges in these pictures that are only apparent.