The diameter of random Cayley digraphs of given degree

Details The diameter of random Cayley digraphs of given degree The diameter of random Cayley digraphs of given degree

2007-06-24

arxiv.org/abs/0706.3539v1

Mathematics

Combinatorics

11 pages

Scientific paper

We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.

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