Random Cayley digraphs of diameter 2 and given degree

Lladser, Manuel E.; Potočnik, Primož; Širáň, Jozef and WIlson, Mark C. (2012). Random Cayley digraphs of diameter 2 and given degree. Discrete Mathematics and Theoretical Computer Science, 14(2) pp. 83–90.

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 → ∞ and k = f (n), focusing on the functions f (n) = ⌊n^δ⌋ and f(n) = ⌊cn⌋. In both instances we show that this probability converges to 1 as n → ∞ for arbitrary fixed δ ∈ (½, 1) and c ∈ (0,½), respectively, with a much larger convergence rate in the second case and with sharper results for Abelian groups.

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