Miniatures

Here are some prototype results which illustrate the relation between the topology of the graph and number theoretical properties:
  1. The graph on Zn generated by T(x) = 2x is connected if and only if n is a power of two.
  2. The graph on Zn* generated by T(x)=2x is connected if and only if n is a power of two or if n is prime and 2 is a primitive root modulo n.
  3. The graph on Zn* generated by T(x)=x2 is connected if and only if n is a Fermat prime.
  4. The graph on Zn generated by T(x) = 3x+1 and S(x) = 2x has exactly 4 triangles if n is prime and larger than 17.
  5. The graph on Zn generated by T(x) = x2 and S(x) = x3 is connected if and only if n is a Pierpont prime.
  6. The graph on Zn generated by T(x) = x2 + a and S(x) = x2 + b is the union of two disjoint isomorphic graphs if n,a,b are even and n is not a multiple of 8.
  7. The graph on Zn generated by T(x)=x2 + a and S(x) = x2 + b is bipartite if n is even and a,b are odd.