Enumerative problems inspired by Mayer's theory of cluster integrals

0.31 MB·English

by  Pierre Leroux

Enumerative problems inspired by
Mayer’s theory of cluster integrals
4 Pierre Leroux, LaCIM, UQAM
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0
December 31, 2003
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n
a
Abstract
J
1 The basic functional equations for connected and 2-connnected graphs can be
traced back to the statistical physicists Mayer and Husimi. They play an essential

] roleinestablishingrigorouslythe virialexpansionforimperfect gases. We surveythis
O
approachand inspired by these equations, we investigate the problem of enumerating
C
some classes of connected graphs all of whose blocks are contained in a given class
h. B. Included are the species of Husimi graphs (B = ”complete graphs”), cacti (B =
t ”unorientedcycles”),andorientedcacti(B =”orientedcycles”). Foreachofthese,we
a
considerthequestionoftheirlabelledorunlabelledenumerationandoftheirmolecular
m
expansion, according (or not) to their block-size distributions.
[

R´esum´e
1
v Les ´equations fonctionnelles de base pour les graphes connexes et 2-connexes
1 remontent aux physiciens Mayer et Husimi. Ces relations sont importantes pour
0 ´etablir rigoureusement le d´eveloppement du viriel pour les gaz imparfaits. Nous
0
pr´esentonscetted´emarcheetinspir´esparcesrelations,nousexaminonsleprobl`emedu
1
d´enombrement de quelques classes de graphes connexes dont les blocs sont pris dans
0
4 une famille B donn´ee. Cela inclut les graphes de Husimi (B = ”graphes complets”),
0 les cactus (B = ”polygones”), et les cactus orient´es (B = ”cycles orient´es”). Pour
/ chacunedecesesp`eces,ons’int´eresseaud´enombrement´etiquet´e,ordinaireouselonla
h
t distributiondestaillesdesblocs,aud´enombrementnon-´etiquet´e,etaud´eveloppement
a
mol´eculaire.
m
:
v
1 Introduction
i
X
r 1.1 Combinatorial species and functional equations for connectedgraphs
a
and blocks
Informally, a combinatorial species is a class of labelled discrete structures which is closed
under isomorphisms induced by relabelling along bijections. See Joyal [13] and Bergeron,
Labelle, Leroux [2] for an introduction to the theory of species. To each species F are
associated a number of series expansions among which are the (exponential) generating
function, F(x), for labelled enumeration, defined by
xn
F(x) = |F[n]| , (1)
n!
n≥0
X
1
where |F[n]| denotes the number of F-structures on the set [n] = {1,2,...,n}, the (ordi-
nary) tilde generating function F(x), for unlabelled enumeration, defined by
F(x) = F xn, (2)
e n
n≥0
X
e e
where F denotes the number of isomorphism classes of F-structures of size n, the cycle
n
index series Z (x ,x ,x ,···), defined by
F 1 2 3
e
1
Z (x ,x ,x ,···) = fixF[σ] xσ1xσ2xσ3··· , (3)
F 1 2 3 n! 1 2 3 
nX≥0 σX∈Sn
 
where S denotes the group of permutations of [n], fixF[σ] is the number of F-structures
n
on [n] left fixed by σ, and σ is the number of cycles of length j in σ, and the molecular
j
expansion of F, which is a description of the F-structures and a classification according
to their stabilizers and will be discussed later.
Combinatorial operations are defined on species: sum, product, (partitional) compo-
sition, derivation, which correspond to the usual operations on the exponential generat-
ing functions. And there are rules for computing the other associated series, involving
plethysm. See [2] for more details. An equality F = G between species is a family of
bijections between structures,
α : F[U] → G[U],
U
where U ranges over all underlying (labelling) sets, which commutes under relabellings.
Such an identity gives rise to an equality between all the series expansion associated to
species.
1 3 6
8
5
7
2
4
Figure 1: A simple graph g and its connected components
For example, the fact that any simple graph on a set (of vertices) U is the disjoint
union of connected simple graphs (see Figure 1) is expressed by the equation
G = E(C), (4)
where G denotes the species of (simple) graphs, C, that of connected graphs, and E, the
species of Sets (in French: Ensembles). There correspond the well-known relations for
their exponential generating functions,
G(x) = exp(C(x)) (5)
2
and for their tilde generating functions,
G(x) = Z (C(x),C(x2),...)
E
1
e = exp e eC(xk) . (6)
 k 
k≥1
X
 e 
Definitions. A cutpoint (or articulation point) of a connected graph g is a vertex of g
whose removal yields a disconnected graph. A connected graph is called 2-connected if
it has no cutpoint. A block in a simple graph is a maximal 2-connected subgraph. The
block-graph of a graph g is a new graph whosevertices are the blocks of g and whose edges
correspond to blocks having a common cutpoint. The block-cutpoint tree of a connected
graph g is a graph whose vertices are the blocks and the cutpoints of g and whose edges
correspond to incidence relations in g. See Figure 2.
a
c d A B
B
A
e
A B
i
e i
b
C C
f
D C h
j D
D
h k
E
g F F
E E
F
p r l
o
H o
G H
H
I
m
s
s
I G
q t n G
I
a) b) c)
Figure 2: a) A connected graph g, b) the block-graph of g, c) the block-cutpoint tree of g
Now let B be a given species of 2-connected graphs. We denote by C the species of
B
connected graphs all of whose blocks are in B, called C -graphs.
B
Examples 1.1. Here are some examples for various choices of B:
1. If B = B , the class of all 2-connected graphs, then C = C, the species of (all)
a B
connected graphs.
2. If B = K , the class of ”edges”, then C = a, the species of (unrooted, free) trees
2 B
a
( for French arbres).
3
3. If B = {P ,m ≥ 2}, where P denotes the class of size-m polygons (by convention,
m m
P = K ), then C = Ca, the species of cacti. A cactus can also be defined as a
2 2 B
connectedgraphinwhichnoedgeliesinmorethanonecycle. Figure3,a),represents
a typical cactus.
i m p i m p
f f
o e o e
a c a c
k k
j g j g
h h
l l
b n d b n d
a) b)
Figure 3: a) a typical cactus, b) a typical oriented cactus
4. If B = K = P , the class of ”triangles”, then C = δ, the class of triangular cacti.
3 3 B
5. If B = {K ,n ≥ 2}, the family of complete graphs, then C = Hu, the species of
n B
Husimi graphs, that is of connected graphs whose blocks are complete graphs. They
were first (informally) introduced by Husimi in [12]. A Husimi graph is shown in
Figure 2, b). See also Figure 7. It can be easily shown that any Husimi graph is the
block-graph of some connected graph.
6. If B = {C ,n ≥ 2}, the family of oriented cycles, then C = Oc, the species of
n B
oriented cacti. Figure 3, b) shows a typical oriented cactus. These structures were
introduced by C. Springer [28] in 1996. Although directed graphs are involved here,
the functional equations (7) and (12) given below are still valid.
Remark. Cacti where first called Husimi trees. See for example [9], [11], [26] and [29].
However this term received much criticism since they are not necessarily trees. Also, a
careful reading of Husimi’s article [12] shows that the graphs he has in mind and that he
enumerates (see formula (42) below) are the Husimi graphs defined in item 5 above. The
term cactus is now widely used. See Harary and Palmer [10]. Cacti appear regularly in
the mathematical litterature, for example in the classification of base matroids [20], and
in combinatorial optimization [4].
The following functional equation (see (7)) is fairly well known. It can be found in
various forms and with varying degrees of generality in [2], [10], [13], [17], [18], [19], [24],
[26], [27]. In fact, it was anticipated by the physicists (see [12] and [29]) in the context
4
of Mayers’ theory of cluster integrals as we will see below. The form given here, in the
structurallanguageofspecies,isthemostgeneralonesincealltheseriesexpansionsfollow.
It is also the easiest form to prove.
Recall that for any species F = F(X), the derivative F′ of F is the species defined as
follows: an F′-structure on a set U is an F-structure on the set U ∪{∗}, where ∗ is an
external (unlabelled) element. In other words, one sets
F′[U] = F[U +{∗}].
Moreover, the operation F 7→ F•, of pointing (or rooting) F-structures at an element of
the underlying set, can be defined by
F• = X ·F′.
Theorem 1.1 Let B be a class of 2-connected graphs and C , the species of connected
B
graphs all of whose blocks are in B. We then have the functional equation
C′ = E(B′(C•)). (7)
B B
Figure 4: C′ = E(B′(C•))
B B
Proof. See Figure 4. 2
Multiplying (7) by X, one finds
C• = X ·E(B′(C•)), (8)
B B
and, for the exponential generating function,
C•(x) = x·exp(B′(C•(x))). (9)
B B
1.2 Weighted versions
Weighted versions of these equations are needed in the applications. See for example
Uhlenbeck and Ford [29]. A weighted species is a species F together with weight functions
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w = w : F[U] → IK defined on F-structures, which commute with the relabellings. Here
U
IK is a commutative ring in which the weights are taken, usually a ring of polynomials
or formal power series over a field of characteristic zero. We write F = F to emphasize
w
the fact that F is a weighted species with weight function w. The associated generating
functions are then adapted by replacing set cardinalities |A| by total weights
|A| = w(a).
w
a∈A
X
Thebasicoperationsonspeciesarealsoadaptedtotheweightedcontext, usingtheconcept
of Cartesian product of weighted sets: Let (A,u) and (B,v) be weighted sets. A weight
function w is defined on the Cartesian product A×B by
w(a,b) = u(a)·v(b).
We then have |A×B| = |A| ·|B| .
w u v
Definition. A weight function w on the species G of graphs is said to be multiplicative
on the connected components if for any graph g ∈G[U], whose connected components are
c ,c ,...,c , we have
1 2 k
w(g) = w(c )w(c )···w(c ).
1 2 k
Examples 1.2. The following weight functions w on the species of graphs are multiplica-
tive on the connected components.
1. w (g) := ye(g), where e(g) is the number of edges of g.
1
2. w (g) = graph complexity of g := number of maximal spanning forests of g.
2
3. w (g) := xn0xn1xn2···, where n is the number of vertices of degree i.
3 0 1 2 i
Theorem 1.2 Let w be a weight function on graphs which is multiplicative on the con-
nected components. Then we have
G = E(C ). (10)
w w
For the exponential generating functions, we have
G (x) = exp(C (x)),
w w
where G (x) = |G[n]| xn = ( w(g))xn, and similarly for C (x).
w n≥0 w n! n≥0 g∈G[n] n! w
Definition. A wPeight function on coPnnectePd graphs is said to be block-multiplicative if
for any connected graph c, whose blocks are b ,b ,...,b , we have
1 2 k
w(c) = w(b )w(b )···w(b ).
1 2 k
Examples 1.3. The weight functions w (g) = ye(g) and w (g) = graph complexity of
1 2
g of Examples 1.2 are block-multiplicative, but not the function w (g) = xn0xn1xn2···.
3 0 1 2
6
Another example of a block-multiplicative weight function is obtained by introducing
formal variables y (i ≥ 2) marking the block sizes. In other terms, if the connected graph
i
c has n blocks of size i, for i = 2,3,..., one sets
i
w(c) = yn2yn3···. (11)
2 3
The following result is then simply the weighted version of Theorem 1.1.
Theorem 1.3. Let w be a block-multiplicative weight function on connected graphs
whose blocks are in a given species B. Then we have
(C•) = X ·E(B′ ((C•) )). (12)
B w w B w
1.3 Outline
In the next section, we see how equations (10) and (12) are involved in the thermodynam-
ical study of imperfect (or non ideal) gases, following Mayers’ theory of cluster integrals
[21], as presented in Uhlenbeck and Ford [29]. In particular, the virial expansion, which is
akindof asymptoticrefinementof theperfectgases law, isestablished rigourously, atleast
in its formal power series form. See equation (34) below. It is amazing to realize that the
coefficients of the virial expansion involve directly the total valuation |B [n]| , for n ≥ 2,
a w
of 2-connected graphs. An important role in this theory is also played by the enumerative
formula (42) for labelled Husimi trees according to their block-size distribution, which
extends Cayley’s formula nn−2 for the number of labelled trees of size n.
Motivated by this, we first consider, in Section 3, the labelled enumeration of some
classesofconnectedgraphsoftheformC ,accordingornottotheirblock-sizedistribution.
B
Included are the species of Husimi graphs, cacti, and oriented cacti. The methods involve
theLagrangeinversionformulaandPru¨fer-typebijections. Itisalsonaturaltoexaminethe
question of unlabelled enumeration of these structures. This is a more difficult problem,
for two reasons. First, equation (12) deals with rooted structures and it is necessary to
introduce a tool for counting the unrooted ones. Traditionally, this is done by extending
the Dissimilarity charactistic formula for trees of Otter [25]. See for example [9]. Inspired
by formulas of Norman ([6], (18)) and Robinson ([27], Theorem 7), we have given over
the years a more structural formula which we call the Dissymmetry theorem for graphs,
whose proof is remarkably simple and which can esily be adapted to various classes of
tree-like structures; see [2], [3], [7], [14], [15], [16], [18], [19]. Second, as for trees, it should
not be expected to obtain simple closed expressions butrather recurrence formulas for the
number of unlabelled C -structures. Three examples are given in that section.
B
Finally, in Section 4, wepresentthemolecular expansion ofsomeof thesespecies. This
expansion can be computed first for the rooted species, by iteration, and the dissymmetry
theorem is then invoked for the unrooted ones. The computations can be carried out
easily using the Maple package ”Devmol” developped at LaCIM and available at the URL
www.lacim.uqam.ca; see [1].
Acknowledgements. This talk is partly taken from my student M´elanie Nadeau’s
”M´emoire de maˆıtrise” [23]. I would like to thank her and Pierre Auger for their consider-
7
able help, and also Abdelmalek Abdesselam, Andr´e Joyal, Gilbert Labelle, Bob Robinson,
and Alan Sokal, for useful discussions.
2 Some statistical mechanics
2.1 Partition functions for the non-ideal gas
Consider a non-ideal gas, formed of N particles interacting in a vessel V ⊆ IR3 (whose
−→ −→ −→
volume is also denoted by V) and whose positions are x ,x ,...,x . The Hamiltonian of
1 2 N
the system is of the form
N −→p 2
i −→ −→ −→
H = +U(x ) + ϕ(|x −x |), (13)
i i j
2m !
i=1 1≤i<j≤N
X X
where −→p is the linear momentum vector and −→pi2 is the kinetic energy of the ith particle,
i 2m
−→ −→ −→ −→
U(x ) is the potential at position x due to outside forces (e.g., walls), |x −x | = r is
i i i j ij
−→ −→
the distance between the particles x and x , and it is assumed that the particles interact
i j
only pairwise through the central potential ϕ(r). This potential function ϕ has a typical
form shown in Figure 5 a).
ϕ f
r0 r1 r r0 r1 r
−1
a) b)
Figure 5: a) the function ϕ(r), b) the function f(r)
The canonical partition function Z(V,N,T) is defined by
1
Z(V,N,T) = exp(−βH)dΓ, (14)
N!h3N
Z
where h is Planck’s constant, β = 1 , T is the absolute temperature and k is Boltzmann’s
kT
−→ −→ −→ −→
constant, and Γ represents the state space x ,...,x ,p ,...,p of dimension 6N. A first
1 N 1 N
−→
simplification comes from the assumption that the potential energy U(x ) is negligible or
i
−→
null. Secondly, the integral over the momenta p in (14) is a productof Gaussian integrals
i
which are easily evaluated so that the canonical partition function can now be written as
1
−→ −→ −→ −→
Z(V,N,T) = ··· exp −β ϕ(|x −x |) dx ···dx , (15)
N!λ3N  i j  1 N
V V
Z Z i<j
X
 
8
where λ = h(2πmkT)−12.
Mathematically, the grand-canonical distribution is simply the generating function for
the canonical partition functions, defined by
∞
Z (V,T,z) = Z(V,N,T)(λ3z)N, (16)
gr
N=0
X
where the variable z is called the fugacity or the activity. All the macroscopic parameters
of the system are then defined in terms of this grand canonical ensemble. For example,
the pressure, P, the average number of particles, N, and the density, ρ, are defined by
P 1 ∂ N
= logZ (V,T,z), N =z logZ (V,T,z), and ρ:= . (17)
gr gr
kT V ∂z V
2.2 The virial expansion
In order to better explain the thermodynamic behaviour of non ideal gases, Kamerlingh
Onnes proposed, in 1901, a series expansion of the form
2 3
P N N N
= +γ (T) +γ (T) +···, (18)
2 3
kT V V V
! !
called the virial expansion. Here γ (T) is the second virial coefficient, γ (T) the third, etc.
2 3
This expansion was first derived theoretically from the partition function Z by Mayer
gr
[21] around 1930. It is the starting point of Mayer’s theory of ”cluster integrals”. Mayer’s
idea consists in setting
−→ −→
1+f = exp(−βϕ(|x −x |)), (19)
ij i j
where f = f(r ). The general form of the function f(r) = exp(−βϕ(r)) −1 is shown
ij ij
in Figure 5, b). In particular, f(r) vanishes when r is greater than the range r of the
1
interaction potential. By substituting in thecanonical partition function (15), one obtains
1
−→ −→
Z(V,N,T) = ··· (1+f )dx ···dx . (20)
N!λ3N ij 1 N
V V
Z Z 1≤i<j≤N
Y
The terms obtained by expanding the product (1+f ) can be represented
1≤i<j≤N ij
by simple graphs where the vertices are the particles and the edges are the chosen factors
Q
f . The partition function (20) can then be rewritten in the form
ij
1
−→ −→
Z(V,N,T) = ··· f dx ···dx
N!λ3N ij 1 N
V V
g∈XG[N]Z Z {i,Yj}∈g
1
= W(g), (21)
N!λ3N
g∈XG[N]
where the weight W(g) of a graph g is given by the integral
−→ −→
W(g) = ··· f dx ···dx . (22)
ij 1 N
V V
Z Z {i,Yj}∈g
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For the grand canonical function, we then have
∞
Z (V,T,z) = Z(V,N,T)(λ3z)N
gr
N=0
X
∞
1
= W(g)(λ3z)N
N!λ3N
NX=0 g∈XG[N]
∞
1
= W(g)zN
N!
NX=0 g∈XG[N]
= G (z). (23)
W
Proposition 2.1 The weight function W is multiplicative on the connected components.
For example, for the graph g of Figure 1, we have
−→ −→
W(g) = f f f f f f f dx ···dx
12 17 27 45 46 56 58 1 8
V8
Z
−→ −→ −→ −→ −→ −→ −→ −→
= f f f dx dx dx dx f f f f dx dx dx dx
12 17 27 1 2 7 3 45 46 56 58 4 5 6 8
Z Z Z
= W(c )W(c )W(c ),
1 2 3
where c , c and c represent the three connected components of g. Following Theorem
1 2 3
1.2, we deduce that
G (z) = exp(C (z)), (24)
W W
where C denotes the weighted species of connected graphs, with
W
zn
C (z) = |C[n]| ,
W W
n!
n≥1
X
and
−→ −→
|C[n]| = ··· f dx ···dx . (25)
W ij 1 n
V V
c∈XC[n]Z Z {iY,j}∈c
Historically, thequantitiesb (V)= 1 |C[n]| arepreciselythecluster integrals ofMayer.
n Vn! W
Equation (24) then provides a combinatorial interpretation for the quantity P . Indeed,
kT
one has, by (17),
P 1
= logZ (V,T,z)
gr
kT V
1
= logG (z)
W
V
1
= C (z). (26)
W
V
10