The number of connected sparsely edged graphs. III. Asymptotic results

Abstract

The number of connected graphs on n labeled points and 9 lines (no loops, no multiple lines) is f(n.9). In the first paper of this series I showed how to find an (increasingly complicated) exact formula for f(n,n+k) for general n and successive k . The method would give an asymptotic approximation t o f(n,n+k) for any fixed k as nw. Here I find this approximation when k = ~( n l '~) , a much more difficult matter. The problem of finding an approximation t o f(n.9) when 9 > n + Cn"3 and ( 2 q l n )log n --w is open.

On the Analysis of Linear Probing Hashing

“…(The quantity γ (n, n − 1) is for instance the number of labeled trees, T n = n n−1 , discussed in Section 1.) The basic problem was first solved by Wright in a famous series of papers [49]- [51]. Wright's solution involves a quadratic recurrent sequence that, after normalization, is the same as that of Section 2, so that the Airy constants make an appearance.…”

Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas

“…(Wright [59] and Bender et al [5] further consider extensions to the case k → ∞, which does not interest us here.) In the form…”

Another proof of Wright's inequalities

“…(1−T ) 3ℓ . Observe that we have W 0 W 0 and W 1 W 1 has been proved in [12]. Now, we can proceed by induction.…”

“…The aim of this note is to provide an alternative and generating function based proof of the inequalities obtained by Sir Wright in [12] . Define recursively b ℓ and c ℓ by…”

“…It has been extensively studied since the Erdős-Rényi's paper [3]. The generating functions associated to the numbers c(n, n + ℓ) are due to Sir E. M. Wright in a series of papers including [11,12]. He also obtained the asymptotic formula for c(n, n + ℓ) for every ℓ = o(n 1/3 ).…”