The diameter of random Cayley digraphs of given degree
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چکیده
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 →∞ and k = f(n). We find a sharp phase transition from 0 to 1 as the order of growth of f(n) increases past √ n log n. In particular, if f(n) is asymptotically linear in n, the probability converges exponentially fast to 1.
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تاریخ انتشار 2006