# Miniatures

Here are some prototype results which illustrate the relation between
the topology of the graph and number theoretical properties:

- The graph on Z
_{n} generated by T(x) = 2x is
connected if and only if n is a power of two.
- The graph on Z
_{n}^{*} 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.
- The graph on Z
_{n}^{*} generated by T(x)=x^{2}
is connected if and only if n is a Fermat prime.
- The graph on Z
_{n} generated by T(x) = 3x+1 and S(x) = 2x
has exactly 4 triangles if n is prime and larger than 17.
- The graph on Z
_{n} generated by T(x) = x^{2} and
S(x) = x^{3} is connected if and only if n is a Pierpont prime.
- The graph on Z
_{n} generated by T(x) = x^{2} + a
and S(x) = x^{2} + b is the union of two disjoint isomorphic
graphs if n,a,b are even and n is not a multiple of 8.
- The graph on Z
_{n} generated by T(x)=x^{2} + a
and S(x) = x^{2} + b is bipartite if n is even and a,b are odd.