Wright’s Constants in Graph Enumeration and Brownian Excursion Area

This is a collection of various results and formulae. The main purpose is to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are no new results. This is an informal note, not intended for publication. 1. Graph enumeration Let C(n, q) be the number of connected graphs with n given (labelled) vertices and q edges. Recall Cayley’s formula C(n, n− 1) = nn−2 for every n ≥ 1. Wright [19] proved that for any fixed k ≥ −1, we have the analoguous asymptotic formula C(n, n + k) ∼ ρkn as n→∞, (1) for some constants ρk given by ρk = 2(1−3k)/2π1/2 Γ(3k/2 + 1) σk, k ≥ −1, (2) with other constants σk given by σ−1 = −1/2, σ0 = 1/4, σ1 = 5/16, and the quadratic recursion relation σk+1 = 3(k + 1) 2 σk + k−1 ∑ j=1 σjσk−j , k ≥ 1. (3) Note the equivalent recursion formula σk+1 = 3k + 2 2 σk + k ∑ j=0 σjσk−j , k ≥ −1. (4) Wright gives in the later paper [20] the same result in the form ρk = 2(1−5k)/23kπ1/2(k − 1)! Γ(3k/2) dk, k ≥ 1, (5) Date: October 3, 2005; revised October 17, 2005. 1