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 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: 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?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] )