The diameters of almost all Cayley digraphs

LetG be a finite group of ordern andS be a subset ofG not containing the identity element ofG. Letp (0<p<1) be a fixed number. We define the set of all labelled Cayley digraphsX(G,S) (S<-G\{1}) ofG as a sample space and assign a probability measure by requiringP(aS)=p for anyaG\{1}. Here it is shown that the probability of the set of Cayley digraphs ofG with diameter 2 approaches 1 as the ordern ofG approaches infinity.This work is supported by the National Natural Science Foundation of China.