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Random Cayley digraphs of diameter 2 and given degree

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.
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Manuel E. Lladser, Primož Potočnik, Jozef Širáň, Mark C. Wilson. Random Cayley digraphs of diameter 2 and given degree. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2012, Vol. 14 no. 2 (2), pp.83--90. ⟨hal-00990591⟩

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