The diameter of a random Cayley graph of Z_q
  Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order q^(1/k).   The same also holds when the generating set is taken to be a symmetric set of size 2k. 
Gideon Amir, Ori Gurel-Gurevich