  Let C(n,k) denote the number of connected graphs with n labeled vertices and
n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to
infinity is known. It has been observed that, if k_n=o(\sqrt{n}), this limit is
asymptotically equal to the $k_n$th moment of the area under the standard
Brownian excursion. These moments have been computed in the literature via
independent methods. In this article we show why this is true for
k_n=o(\sqrt[3]{n}) starting from an observation made by Joel Spencer. The
elementary argument uses a result about strong embedding of the Uniform
empirical process in the Brownian bridge proved by Komlos, Major, and Tusnady.
