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Preprint submitted on 12 Jun 2007
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ANOTHER PROOF OF WRIGHT’S INEQUALITIES
Vlady Ravelomanana
To cite this version:
Vlady Ravelomanana. ANOTHER PROOF OF WRIGHT’S INEQUALITIES. 2007. �hal-00153936�
hal-00153936, version 1 - 12 Jun 2007
VLADY RAVELOMANANA
Abstract. 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 withℓedges more than vertices.
1. Preliminaries Forn ≥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 Erd˝os-R´enyi’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(n1/3). 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.
Forℓ ≥ −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) = P∞
n=0c(n, n+ℓ)zn!n. Let T(z) be the EGF of the Cayley’s rooted labeled trees. It is well known that T(z) = z eT(z) = P
n≥1nn−1zn!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 [zn]A(z)≤[zn]B(z) then we denote this relation AB orA(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 = 245 and c1 = 1924. Define recursively bℓ and cℓ by (1) 2(ℓ+ 1)bℓ+1 = 3ℓ(ℓ+ 1)bℓ+ 3
ℓ−1
X
t=1
t(ℓ−t)btbℓ−t, (ℓ≥1) and
2(3ℓ+ 2)cℓ+1 = 8(ℓ+ 1)bℓ+1+ 3ℓbℓ+ (3ℓ+ 2)(3ℓ−1)cℓ
+ 6
ℓ−1
X
t=1
t(3ℓ−3t−1)btcℓ−t, (ℓ≥1) . (2)
Then, for all ℓ ≥1
(3) bℓ
(1−T(z))3ℓ − cℓ
(1−T(z))3ℓ−1 Wℓ(z) bℓ
(1−T(z))3ℓ .
(3) is known as Wright’s inequalities and such results has been extremely useful in the enumerative study of graphs as well as in the theory of random graphs [2, 5, 6, 7, 10].
Our proof of (3) is based upon two ingredients:
Fact 1. We know that the EGFs Wℓ satisfy W−1 =T − T22, W0 =−12log (1−T)− T2 − T42 and
(4)
(1−T)ϑzWℓ+1+ (ℓ+ 1) Wℓ+1 =
ϑ2z−3ϑz
2 −ℓ
Wℓ+ 1 2
ℓ
X
k=0
(ϑzWk) (ϑzWℓ−k) , (ℓ ≥0), where T = T(z), Wk = Wk(z) and ϑz =z∂z∂ corresponds to marking a vertex (such combi- natorial operator consists to choose a vertex among the others). For the combinatorial sense of (4), we refer the reader to [1, 5] or [11].
Fact 2. Let A and B be two formal power series and ℓ ∈ N. If (1−T)ϑzA+ (ℓ+ 1)A (1−T)ϑzB+ (ℓ+ 1)B then AB.
To prove Fact 2, fix ℓ≥0. We write (5) B(z)−A(z) =
∞
X
n=0
(bn−an)zn
n! and ∀n, cn =bn−an. Suppose that (1−T)ϑzA+ (ℓ+ 1)A(1−T)ϑzB + (ℓ+ 1)B. We then have
n! [zn] ((1−T(z))ϑz(B(z)−A(z)) + (ℓ+ 1) (B(z)−A(z))) = (n+ℓ+ 1)cn−
n
X
k=1
n k
kk−1(n−k)cn−k≥0. (6)
It is now easily seen that ∀n, cn≥0. Therefore, AB.
Our proof of (3) is divided into two parts each of each are given in the next Sections.
2. Proof of Wℓ bℓ
(1−T)3ℓ
Define the family (Wℓ)ℓ≥0 as W0 =−1
2log (1−T) and for ℓ∈N⋆, Wℓ = (1−Tbℓ)3ℓ. Observe that we have W0 W0 and W1 W1 has been proved in [12]. Now, we can proceed by induction. Suppose that for 2 ≤ i ≤ ℓ, Wi Wi = bi
(1−T)3i and let us prove that Wℓ+1 Wℓ+1 = (1−Tbℓ+1)3ℓ+3. Simple calculations show that
(7)
ϑ2z−ϑz
2
Wℓ ϑ2z
2 Wℓ 3ℓ(3ℓ+ 2) 2
bℓ
(1−T)3ℓ+4 −3ℓ(3ℓ+ 2) 2
bℓ
(1−T)3ℓ+3,
(8) ϑzW0
ϑzWℓ
3ℓbℓ
2
bℓ
(1−T)3ℓ+4 −3ℓbℓ
2
bℓ
(1−T)3ℓ+3 and
(9) 1 2
ℓ−1
X
p=1
ϑzWp
ϑzWℓ−p
1 2
ℓ−1
X
p=1
9p(ℓ−p)bpbℓ−p
! 1
(1−T)3ℓ+4 − 1 (1−T)3ℓ+3
.
Summing (7), (8), (9), using the recurrence (1) and the induction hypothesis, we find that (10) (1−T)ϑzWℓ+1+ (ℓ+ 1)Wℓ+1 3(ℓ+ 1)bℓ+1
(1−T)3ℓ+4 −3(ℓ+ 1)bℓ+1
(1−T)3ℓ+3 . Since
(11) (1−T)ϑzWℓ+1+ (ℓ+ 1)Wℓ+1 = 3(ℓ+ 1)bℓ+1
(1−T)3ℓ+4 − 2(ℓ+ 1)bℓ+1 (1−T)3ℓ+3 by Fact 2, we have Wℓ+1 Wℓ+1.
3. Proof of (1−Tbℓ)3ℓ − (1−Tcℓ)3ℓ−1 Wℓ
Define W0 = W0 and for ℓ ∈ N⋆, Wℓ = (1−Tbℓ)3ℓ − cℓ
(1−T)3ℓ−1. As before, we shall proceed by induction. We have W0 W0 and
(12) W1− W1 = 13
12 (1−T) −1 2 − T
8 + T2 24 13
12
1
(1−T)−T −1
= 13T2
12(1−T) 0. Suppose that for 2 ≤ k ≤ ℓ, Wk = (1−Tbk)3k − ck
(1−T)3k−1 Wk. We have to prove that Wℓ+1 = (1−Tbℓ+1)3ℓ+3 − cℓ+1
(1−T)3ℓ+2 Wℓ+1. For this purpose, define Ψℓ+1 as Ψℓ+1 =
ϑ2z−3ϑz
2 −ℓ
Wℓ+ (ϑzW0) (ϑzWℓ) + 1 2
ℓ−1
X
k=1
ϑzWk− (3ℓ−1)cℓ
(1−T)3ℓ
ϑzWℓ−k
−
αℓ
(1−T)3ℓ+2 + βℓ
(1−T)3ℓ+1 + γℓ
(1−T)3ℓ + δℓ
(1−T)3ℓ−1
, (13)
4 VLADY RAVELOMANANA
where αℓ,βℓ, γℓ and δℓ are given by αℓ = (7ℓ+ 4)cℓ+1
2 −3(ℓ+ 1)bℓ+1−3
4ℓbℓ+(3ℓ−1)(3ℓ+ 4)
4 cℓ
+ 1 2
ℓ−1
X
t=1
(3t−1)ct(3ℓ−3t−1)cℓ−t, (14)
βℓ = −(3ℓ+ 2)cℓ+1
2 + 2(ℓ+ 1)bℓ+1− 3
4ℓbℓ− (3ℓ−1)(3ℓ+ 4)
4 cℓ
− 1 2
ℓ−1
X
t=1
(3t−1)ct(3ℓ−3t−1)cℓ−t, (15)
γℓ = ℓbℓ
2 +(3ℓ−1)cℓ
2 and δℓ=−ℓ−1 2 cℓ . (16)
Rewritting the formal power series (1−Tα)ℓ3ℓ+2 + (1−Tβ)ℓ3ℓ+1 + (1−Tγℓ)3ℓ +(1−Tδℓ)3ℓ−1 as follows (7ℓ+ 4)/2cℓ+1−3(ℓ+ 1)bℓ+1−3/4ℓbℓ
(1−T)3ℓ+2 − (3ℓ+ 2)/2cℓ+1−2(ℓ+ 1)bℓ+1+ 3/4ℓbℓ
(1−T)3ℓ+1 + (3ℓ−1)(3ℓ+ 4)cℓ
1
(1−T)3ℓ+2 − 1 (1−T)3ℓ+1
+ 2ℓbℓ 2(1−T)3ℓ +
(3ℓ−1)cℓ
2(1−T)3ℓ − (ℓ−1)cℓ 2(1−T)3ℓ−1
, (17)
it is easily seen that if the quantity (coming from the denominators of the 2 first terms of the above equation)
(2ℓ+ 1)cℓ+1−(ℓ+ 1)bℓ+1−3
2ℓbℓ ≥0 (18)
then (1−Tα)ℓ3ℓ+2 + (1−Tβℓ)3ℓ+1 + (1−Tγℓ)3ℓ + (1−Tδℓ)3ℓ
−1 0. (We used 1/(1−T)a 1/(1−T)b if a≥b).
Using (1) and (2), after simple algebra we have (18). Therefore by construction, RHS of(4) Ψℓ+1. After nice cancellations, it yields
(19) Ψℓ+1 = 3(ℓ+ 1)bℓ+1
(1−T)3ℓ+4 −2(ℓ+ 1)bℓ+1+ (3ℓ+ 2)cℓ+1
(1−T)3ℓ+3 + (2ℓ+ 1)cℓ+1
(1−T)3ℓ+2 .
Remarking that (1−T)ϑzWℓ+1 + (ℓ+ 1) Wℓ+1 = Ψℓ+1, we have completed the proof of Wℓ+1 Wℓ+1.
References
[1] Bender, E. A., Canfield, E. R. and McKay B. D. (1990). The asymptotic number of labelled connected graphs with a given number of vertices and edges.Random Structures and Algorithms, 1:127–169.
[2] Bollob´as, B. (1985).Random Graphs. Academic Press, London.
[3] Erd˝os, P. and R´enyi A. (1959). On random graphs.Publ. Math. Debrecen, 6:290–297.
[4] Flajolet, P. and Sedgewick, R.Analytic Combinatorics.To appear (chapters are avalaible as Inria research reports). Seehttp://algo.inria.f r/f lajolet/P ublications/books.html.
[5] Janson, S., Knuth, D. E., Luczak, T. and Pittel B. (1993). The birth of the giant component.Random Structures and Algorithms, 4:233–358.
[6] Janson, S., Luczak, T. and Ruci´nski A. (2000).Random Graphs. John Wiley, New York.
[7] Luczak, T. (1990). On the number of sparse connected graphs. Random Structures and Algorithms, 1:171–174.
[8] Pittel, B. and Wormald, N. C. (2005). Counting connected graphs inside out.J. Combinatorial Th. Ser.
B, 93: 127–172.
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[10] Ravelomanana (2006). The Average Size of Giant Components between the Double-Jump.Algorithmica, 46: 529–555.
[11] Wright, E. M. (1977). The Number of Connected Sparsely Edged Graphs. Journal of Graph Theory, 1:317–330.
[12] Wright, E. M. (1980). The Number of Connected Sparsely Edged Graphs III: Asymptotic results.Journal of Graph Theory, 4:393–407.
E-mail address: [email protected]
Vlady RAVELOMANANA, LIPN – UMR 7030, Institut Galil´ee –Universit´e de Paris-Nord, 99, Avenue J. B. Cl´ement. F 93430 Villetaneuse, France.
