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
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?
