Another Proof of Wright's Inequalities

We present a short way of proving the inequalities obtained by Wright in [Journal of Graph Theory, 4: 393 – 407 (1980)] concerning the number of connected graphs with l edges more than vertices. 1. Preliminaries For n ≥ 0 and −1 ≤ l ≤ ( n 2 ) − n, let c(n, n + l) be the number of connected graphs with n vertices and n+ l edges. Quantifying c(n, n+ l) represents one of the fundamental tasks in the theory of random graphs. 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). Using different methods, Bender, Canfield and McKay [1], Pittel and Wormald [8] and van der Hofstad and Spencer [9] were able to determine the asymptotic value of c(n, n+ l) for all ranges of n and l. For l ≥ −1, let Wl be the exponential generating function (EGF, for short) of the family of connected graphs with n vertices and n + l edges. Thus, Wl(z) = ∑∞ n=0 c(n, n + l) z n! . Let T (z) be the EGF of the Cayley’s rooted labeled trees. It is well known that T (z) = z e (z) = ∑ n≥1 n n−1 z n! (see for example [4, 5]). Among other results, Wright proved that the functions Wl(z), l ≥ −1, can be expressed in terms of T (z). Such results allowed penetrating and precise analysis when studying random graphs processes as it has been shown for example in the giant paper [5]. Throughout the rest of this note, all formal power series are univariate. Therefore, for sake of simplicity we will often omit the variable z so that T ≡ T (z), Wi ≡ Wi(z) and so on. We need the following notations. Definition. If A and B are two formal power series such that for all n ≥ 0 we have [z]A(z) ≤ [z]B(z) then we denote this relation A B or A(z) B(z). The aim of this note is to provide an alternative and generating function based proof of the inequalities obtained by Sir Wright in [12] (in particular, he used numerous intermediate lemmas). More precisely, Wright obtained the following. 1 2 VLADY RAVELOMANANA Theorem (Wright 1980). Let b1 = 5 24 and c1 = 19 24 . Define recursively bl and cl by (1) 2(l+ 1)bl+1 = 3l(l+ 1)bl + 3 l−1