Brownian approximation to counting graphs

Details Brownian approximation to counting graphs Brownian approximation to counting graphs

2012-01-27

arxiv.org/abs/1201.5909v1

Mathematics

Probability

8 pages

Scientific paper

Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. We show that when k=o(\sqrt{n}/\ln n), and n tends to infinity, then, when properly scaled, C(n,k) is asymptotically equal to the k-th moment of the area under a standard Brownian excursion. The elementary (re)proof uses a result about strong embedding of the Uniform empirical process in the Brownian bridge proved by Komlos, Major, and Tusnady.

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