Figure 2.1: The block structure of a graph.
may further need counting formulas for 2-connected graphs and 3-connected graphs. Having complete set of such identities we can determine the exact numbers using Taylor series expansions atx= 0. To determine the asymptotic number we think of the generating functions as complex valued functions and apply singularity analysis presented in Section 2.4.
2.3 Enumeration of unlabeled planar structures
In the enumeration ofunlabeled graphs cycle index sums introduced by Pólya and Burnside’s lemma play essential roles [76, 118]. To determine the number of unlabeled graphs, the problem is reformulated, so that the answer can be obtained by finding the number of orbits of the appropriate permutation group.
Burnside’s lemma can then be used to express the number of orbits in terms of the number of objects fixed by permutations in the group. Pólya’s enumeration theorem incorporates Burnside’s lemma in terms of an appropriate cycle index and a polynomial called figure counting series. The results in this section are based on the book by Harary and Palmer [76].
Cycle index of a permutation group. LetAbe a group of permutations on object setX={1,· · ·, n}. Note that each permutationσ∈Acan be written uniquely as a product of disjoint cycles. For each integerkfrom 1 tonletik(σ) denote the number of cycles of length k in the disjoint cycle decomposition of σ. The cycle indexZ(A) ofAis a polynomial in the formal variabless1,· · · , sn defined by
Z(A) :=Z(A;s1,· · ·, sn) := 1
|A|
X
σ∈A n
Y
k=1
sikk(σ). For example the cycle index of all the symmetric group is
X
n≥0
Z(Sn) = exp
X
k≥1
sk
k
. (2.3.1)
For convenience we takeZ(S0) = 1.
22 CHAPTER 2. PLANAR STRUCTURES
Burnside’s lemma. Before stating Burnside’s lemma let us recall facts on a permutation group. Let A be a permutation group on object set X = {1,2,· · ·, n}. We say that x and y in X are similar if there is a permuta-tionσ∈Asuch thatσx=y. This is an equivalent relation and the equivalent classes are called theorbitsofA. For eachx∈Xthe setA(x) ={σ∈A|σx=x}
is called a stabilizer of x. If x and y in X are similar, then |A(x)| =|A(y)|. Further for any elementyof an orbit Y ofA,|A|=|A(y)||Y|.
Lemma 2.3.1(Burnside’s lemma). The numberN(A)of orbits ofA satisfies N(A) = 1
|A|
X
σ∈A
i1(σ).
Consider the graph Gin Figure 2.2 and denote by Γ(G) its automorphism group. Then Γ(G) consists of the following four permutations
σ1 = (1)(2)(3)(4)(5)(6) σ2 = (1)(23)(4)(5)(6) σ3 = (1)(2)(3)(4)(56) σ4 = (1)(23)(4)(56),
andi1(σ1) = 6,i1(σ2) = 4, i1(σ3) = 4, andi1(σ4) = 2. Thus 1
|Γ(G)| X
σ∈Γ(G)
i1(σ) =1
4(6 + 4 + 4 + 2) = 4. Obviously there are four orbits of Γ(G): {1},{2,3},{4}, and{5,6}.
Figure 2.2: A graph with two fixed points and four orbits.
We may sometimes restrictAto a subsetY ofX, whereY is a union of orbits ofA. We denote byA|Y the set of permutations onY obtained by restricting those ofAtoY. For eachσ∈A, we denote byi1(σ|Y) the number of elements in Y fixed byσ. Then we obtain a restricted form of Burnside’s lemma saying that the numberN(A|Y) of orbits of Arestricted toY satisfies
N(A|Y) = 1
|A|
X
σ∈A
i1(σ|Y). (2.3.2)
2.3. ENUMERATION OF UNLABELED PLANAR STRUCTURES 23
Pólya’s theorems. Let A be a permutation group on object set X = [n] and let I be an identity group on a countable object set Y with at least two elements. The power group IA is the collection YX of functions from X into Y as its object sets. The permutations ofIAconsist of all ordered pairs (σ,id) of σ ∈ A and id ∈ I. The image of any function f in YX under (σ,id) is given by (σ,id)f(x) =f(σx) for eachx∈X, considering thatIA acts onYX. Let ω : Y → {0,1,· · · ,} be a weight function such that ω−1(k) < ∞ for all k= 0,1,· · · and letck =|ω−1(k)|be the number of figures with weight k. The formal power seires in the variablex, defined by
c(x) =X
k≥0
ckxk,
enumerates the elements ofY by weight and is calledfigure counting series.
The weight of a function inYX is defined by ω(f) =X
x∈X
ω(f(x)). (2.3.3)
Thus functions in the same orbit of the power groupIA have the same weight.
LetCk be the number of orbits of weight k. The formal power seriesC(x) = P
k≥0Ckxk is called the configuration counting series or the ordinary gener-ating function with counting sequence {Ck}k. The following Pólya’s theorem expressesC(x) in terms ofZ(A) andc(x).
Theorem 2.3.1. The configuration counting series is obtained by replacing each variablesk in Z(A)by the figure counting seriesc(xk), which we denote by
C(x) =Z(A, c(x)) :=Z(A;c(x), c(x2),· · ·, c(xn)).
For illustration let us count the number of unlabeled pentagon whose vertices are coloured either red or blue. LetX ={1,2,· · ·,5}andY ={red,blue}. Each functionf fromXtoY corresponds to a labeled pentagon with coloured vertices where the vertex labeled withxhas colourf(x). Thus the pentagon represented byf has f−1(red) vertices coloured red and f−1(blue) vertices coloured blue.
We now consider the identity group I acting on Y. To determine the number of unlabeled pentagons whose vertices are coloured either red or blue we should identify the pentagons when one differs from the other only by a rotation or re-flection of the pentagon, that is, we should equip the pentagon with the dihedral group of degree 5, denoted byD5. To remove the labels we should identify two labeled pentagons with coloured vertices whenever their corresponding functions are in the same orbit ofID5. We define the weight functionω :Y → {0,1} by ω(red) = 0 andω(blue) = 1. Then 1 +xis the figure counting series forY and a function of weightk represents a pentagon with 5−kred vertices andkblue vertices. Hence the configuration counting series C(x) = P
k≥0Ckxk counts the number of unlabeled pentagons, where the coefficient Ck is the number of unlabeled pentagons withkblue vertices. From Theorem 2.3.1 we have that
C(x) =Z(D5,1 +x). But it is known that
Z(D5) =s51+ 4s5
10 +s1s22 2 ,
24 CHAPTER 2. PLANAR STRUCTURES
and therefore the ordinary generating function for the counting sequence of the number of unlabeled pentagon whose vertices are coloured either red or blue is
C(x) =Z(D5,1 +x) = (1 +x)5+ 4(1 +x5)
10 +(1 +x)(1 +x2)2 2
= 1 +x+ 2x2+ 2x3+x4+x5, as we can see in Figure 2.3.
Figure 2.3: Pentagon coloured with two colours.
Next let us consider the composition of two permutation groups. Let A and B be permutation groups with objects sets X = {x1, x2,· · ·, xn} and Y = {y1, y2,· · · , ym}. The composition of A with B, denoted by A[B], has object set X×Y and is defined as follows. For eachσ∈Aand each sequence β1, β2,· · ·, βk ofkpermutations inB, there is a permutation inA[B], denoted by [σ;β1, β2,· · · , βk], such that for every ordered pair (xi, yj)∈X×Y,
[σ;β1, β2,· · ·, βk](xi, yj) = (σxi, βjyj).
The following Pólya’s composition theorem shows that the cycle index of the composition ofA withB is obtained by substituting the cycle index ofB into the cycle index ofA.
Theorem 2.3.2. The cycle index Z(A[B]) of the composition of A with B is the polynomial obtained from Z(A) by replacing each variable sk in Z(A) by Z(B;sk, s2k, s3k,· · ·), which is denoted byZ(A)[Z(B)].
Cycle index for a graph. For a graph Gon nvertices with the automor-phism group Γ(G), we writeZ(G) :=Z(Γ(G)), and for a set of graphsC, we writeZ(C) for thecycle index sum forC defined by
Z(C) :=Z(C;s1,· · ·, sn) :=X
G∈C
Z(G;s1,· · ·, sn). (2.3.4) As shown in [18], if ¯C is the set of graphs of C equipped with distinct labels, then
Z(C) =X
n≥0
1 n!
X
G∈C¯n
X
σ∈Γ(G) n
Y
k=1
sikk(σ),
2.3. ENUMERATION OF UNLABELED PLANAR STRUCTURES 25 which coincides with (2.3.4) and shows the close relationship of cycle index sums to exponential generating functions in labeled counting.
The composition of graphs corresponds to the composition of the associated cycle indices. Consider an object setX ={1,· · ·, n}and a permutation group A on X. A composition of n graphs from C is a function f : X → C. Two compositions f and g are similar, f ∼g, if there exists a permutation σ ∈A withf◦σ=g.
Theorem 2.3.3. We writeG for the set of equivalence classes of compositions ofn graphs fromC (with respect to the equivalence relation∼). Then
Z(G) =Z(A) [Z(C)] :=Z(A;Z(C;s1, s2,· · ·), Z(C;s2, s4,· · ·),· · ·), (2.3.5) that is,Z(G)is obtained from Z(A)by replacing each si by
Z(C;si, s2i,· · ·) =X
G∈C
Z(G;si, s2i,· · ·).
Hence, (2.3.5) makes it possible to derive the cycle index sum for a class of graphs by decomposing the graphs into simpler structures with known cycle index sum.
In many cases, such a decomposition is only possible when, for example, one vertex is distinguished from the others in the graphs. A graph with a distin-guished vertex is called a vertex-rooted graph. The automorphism group of a vertex-rooted graph consists of all permutations of the group of the unrooted graph that fix the root vertex. Hence, one can expect a close relation between the cycle indices of unrooted graphs and the cycle indices of their rooted coun-terparts. As shown in [76], ifGis an unlabeled set of graphs and ˆGis the set of graphs inGrooted at a vertex, then
Z( ˆG) =s1
∂
∂s1
Z(G). (2.3.6)
This relationship can be inverted to express the cycle index sum for the unrooted graphs in terms of the cycle index sum for the rooted graphs,
Z(G) =Z s1 0
1
s1Z( ˆG)ds1+Z(G)|s1=0. (2.3.7) Observe that permutations without fixed points are not counted by the cycle indices of the rooted graphs, so that their cycle indices are added as a boundary term toZ(G).
Ordinary generating functions. Once the cycle index sumZ(G) for a class Gof graphs of interest is known, the corresponding ordinary generating function can be derived by replacing the formal variablessi in the cycle index sums by xi. For we know that for a graphG
Z G;x, x2,· · ·
=x|G|. (2.3.8)
Thus letting gn be the number of graphs G ∈ G of given size n, the ordinary generating function forG defined by
G(x) :=X
n≤0
gnxn =X
G∈G
x|G| (2.3.9)
26 CHAPTER 2. PLANAR STRUCTURES
is obtained from the cycle index sum by G(x) = X
G∈G
Z G;x, x2,· · ·
. (2.3.10)
More generally, for a group Aand an ordinary generating functionf(x) we define
Z(A;f(x)) :=Z A;f(x), f(x2), f(x3),· · ·
as the ordinary generating function obtained by substituting each si in Z(A) byf(xi),i≥1.
Once we have ordinary generating functions for graphs we can compute the exact numbers using Taylor series expansions at x = 0 and the asymptotic number using singularity analysis.