Manuel Lladser, Primoz Potocnik, Jana Siagova, Jozef Siran and Mark C. Wilson, submitted to Random Structures and Algorithms, June 2006 (12 pages)
Abstract: We consider random Cayley digraphs of order $latex n$ with uniformly distributed generating set of size $latex k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $latex ntoinfty$ and $latex k=f(n)$. We find a sharp phase transition from 0 to 1 as the order of growth of $latex f(n)$ increases past $latex sqrt{n log n}$. In particular, if $latex f(n)$ is asymptotically linear in $latex n$, the probability converges exponentially fast to 1.