Random Cayley digraphs of diameter 2 and given degree
Manuel E. Lladser (1), Primož Potočnik (2), Jozef Širáň (3) (4), Mark C. Wilson (5)
(1) Department of Applied Mathematics [Boulder]
(2) Faculty of Mathematics and Physics [Ljubljana]
(3) School of Mathematics and Statistics [Milton Keynes]
(4) Department of Mathematics and Descriptive Geometry [Bratislava]
(5) Department of Computer Science [Auckland]
Abstract
We consider random Cayley digraphs of order n with uniformly distributed generating sets of size k. Specifically, we are interested in the asymptotics of the probability that such a Cayley digraph has diameter two as n -> infinity and k = f(n), focusing on the functions f(n) = left perpendicularn(delta)right perpendicular and f(n) = left perpendicularcnright perpendicular. In both instances we show that this probability converges to 1 as n -> infinity for arbitrary fixed delta is an element of (1/2, 1) and c is an element of (0, 1/2), respectively, with a much larger convergence rate in the second case and with sharper results for Abelian groups.
