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Counting Prime Graphs and Point-Determining Graphs
Using Combinatorial Theory of Species
A Dissertation
Presented to
The Faculty of the Graduate School of Arts and Sciences
Brandeis University
Department of Mathematics
Ira Gessel, Advisor
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
by
Ji Li
August, 2007
The signed version of this signature page is on file at the Graduate School of Arts
and Sciences at Brandeis University.
This dissertation, directed and approved by Ji Li’s committee, has been accepted
and approved by the Faculty of Brandeis University in partial fulfillment of the
requirements for the degree of:
DOCTOR OF PHILOSOPHY
Adam Jaffe, Dean of Arts and Sciences
Dissertation Committee:
Ira Gessel, Dept. of Mathematics, Chair.
Susan Parker, Dept. of Mathematics
Alex Postnikov, Dept. of Mathematics, MIT
c© Copyright by
Ji Li
2007
Dedication
To Kailang Li and Youzhi Jiang.
Lu Man Man Qı Xiu Yuan Xi,
Wu Jiang Shang Xia Er Qiu Suo.
by: Qu Yuan
The way is long with obstacles,
I’m questing for the truth to and fro.
iv
Acknowledgments
I would like to thank my advisor, Professor Ira Gessel, for his constant help and
support, without which this work would never have existed.
I am grateful to the members of my dissertation defense committee, Professor
Susan Parker from Brandeis University and Professor Alex Postnikov from MIT.
I also owe thanks to the faculty, to my fellow students, and to the kind and
supportive staff of the Brandeis Mathematics Department.
v
Abstract
Counting Prime Graphs and Point-Determining Graphs Using
Combinatorial Theory of Species
A dissertation presented to the Faculty of theGraduate School of Arts and Sciences of Brandeis
University, Waltham, Massachusetts
by Ji Li
In this thesis, we enumerate different types of graphs in light of the combinatorial
theory of species initiated by Joyal [13]. The prime graphs with respect to the
Cartesian multiplication is enumerated using the exponential composition of species,
which is constructed based on the arithmetic product of species studied by Maia
and Mendez [18]. The point-determining graphs and bi-point-determining graphs are
enumerated by finding functional equations relating them to the species of graphs.
vi
Contents
List of Figures ix
Chapter 1. Groups Actions and Polya’s Cycle Index Polynomial 1
1.1. Symmetric Groups and Group Actions 1
1.2. Polya’s Cycle Index Polynomials 5
1.3. Exponentiation Group 8
Chapter 2. Combinatorial Theory of Species 15
2.1. Definition of Species 15
2.2. Species Operations 21
2.3. Molecular Species 25
2.4. Compositional Inverse of E+ 31
2.5. Multisort Species 35
2.6. Arithmetic Product of Species 38
Chapter 3. Cartesian Product of Graphs and Prime Graphs 48
3.0. Introduction 48
3.1. Cartesian Product of Graphs 49
3.2. Labeled Prime Graphs 54
3.3. Unlabeled Prime Graphs 61
3.4. Exponential Composition with a Molecular Species 67
3.5. Exponential Composition 78
vii
3.6. Cycle Index of Prime Graphs 82
Chapter 4. Point-Determining Graphs 86
4.0. Introduction 86
4.1. Point-Determining Graphs and Co-Point-Determining Graphs 87
4.2. Connected Point-Determining Graphs and Connected Co-Point-
Determining Graphs 91
4.3. Bi-Point-Determining Graphs 96
4.4. 2-Colored Graphs and Connected 2-colored Graphs 112
4.5. Point-Determining 2-Colored Graphs 117
Appendix A. Index of Species 122
Appendix B. General Notations 124
Appendix. Bibliography 127
viii
List of Figures
1.1 Group action of A > B. 3
1.2 Group action of A × B. 3
1.3 Group action of B ≀ A. 4
1.4 Group action of BA. 9
1.5 An element of BA. 10
1.6 Cycle type of an element of BA. 10
2.1 Transport of G -structures. 16
2.2 Species associated to a graph. 17
2.3 A linear order and a permutation. 20
2.4 Addition of species. 21
2.5 Multiplication of species. 23
2.6 Composition of species. 23
2.7 The formula (Xm/A) · (Xn/B) = Xm+n/(A × B). 31
2.8 The formula (Xm/A) (Xn/B) = Xmn/(B ≀ A). 31
2.9 Unlabeled connected graphs. 35
2.10 Multisort species. 37
2.11 The 2-sort species of rooted trees. 38
2.12 A rectangle. 39
ix
2.13 Another rectangle. 39
2.14 A 3-rectangle. 40
2.15 Arithmetic product of species. 42
2.16 Unit of the arithmetic product. 42
2.17 The species E2 ⊡ E2. 43
2.18 The formula (Xm/A) ⊡ (Xn/B) = Xmn/(A × B). 44
2.19 The E2 ⊡ E2-structures. 45
2.20 An element of A × B. 47
3.1 The Cartesian product of graphs. 50
3.2 A prime decomposition. 51
3.3 The automorphism group of a prime power. 56
3.4 The species E2〈E2〉. 69
3.5 The formula ((Xn/B)⊡m)/A = Xnm
/BA. 71
3.6 Unlabeled prime graphs. 84
3.7 Unlabeled non-prime graphs. 85
4.1 Labeled point-determining graphs and co-point-determining graphs. 88
4.2 Transform a graph into a point-determining graph. 89
4.3 Unlabeled point-determining graphs. 91
4.4 Unlabeled connected point-determining graphs and unlabeled connected
co-point-determining graphs. 95
4.5 A phylogenetic tree. 96
4.6 A functional equation for the species of phylogenetic trees. 97
x
4.7 Unlabeled phylogenetic trees. 98
4.8 Unlabeled phylogenetic trees corresponding to the species XE 22 . 99
4.9 An alternating phylogenetic tree. 100
4.10 An illustration of (g, h). 101
4.11 Operations OQ and OP . 103
4.12 An illustration of (g′, h′). 103
4.13 Alternating applications of operations OP and OQ. 105
4.14 Construct a graph from a given triple (π, ϕ, γ). 107
4.15 Unlabeled bi-point-determining graphs. 109
4.16 The 2-sort species of 2-colored graphs. 112
4.17 Unlabeled 2-colored graphs counted by the number of edges. 114
4.18 Unlabeled 2-colored graphs. 115
4.19 Unlabeled connected 2-colored graphs. 116
4.20 The formula Ps(X, Y ) = (1 + X)(1 + Y ) E (Pc≥2(X, Y )). 118
4.21 The formula P(X, Y ) = (1 + X + Y ) E (Pc≥2(X, Y )). 118
4.22 Unlabeled point-determining 2-colored graphs. 120
4.23 Unlabeled point-determining 2-colored graphs with 5 vertices. 121
xi
CHAPTER 1
Groups Actions and Polya’s Cycle Index Polynomial
1.1. Symmetric Groups and Group Actions
The symmetric group of order n, denoted Sn, is the group of permutations of [n] =
1, 2, . . . , n. Let λ = (λ1, λ2, . . . ), where the λi are arranged in weakly decreasing
order, be a partition of n, denoted λ ⊢ n. That is, |λ| =∑
i λi = n. Let σ be
a permutation of [n]. We say λ is the cycle type of σ, denoted by λ = c. t.(σ), if
the λi are the lengths of the cycles in the decomposition of σ into disjoint cycles in
weakly decreasing order. Sometimes we write λ = (1c1(λ), 2c2(λ), . . . ), where ck(λ) is
the number of parts of length k in λ for k ≥ 1.
Example 1.1.1. Let σ =(
14
22
35
41
56
63
)= (3, 5, 6)(1, 4)(2). Then the cycle type of σ is
(3, 2, 1), and c1(σ) = c2(σ) = c3(σ) = 1.
It is well-known that the number of permutations of [n] of cycle type
λ = (1c1 , 2c2, . . . , kck)
is
dλ :=n!
c1! 1c1 c2! 2c2 · · · ck! kck,
in which the denominator
zλ := c1! 1c1 c2! 2c2 · · · ck! kck
1
CHAPTER 1. GROUP ACTIONS
is the number of permutations in Sn that commute with a permutation of cycle type
λ.
Definition 1.1.2. An action of a group A on a set S is a function
ρ : A × S → S,
where for x ∈ A and s ∈ S, ρ(x, s) is written x · s. We say this action is natural if
both of the following conditions are satisfied:
x · (y · s) = (xy) · s, idA ·s = s,
for any x, y ∈ A and s ∈ S.
Let A be a subgroup of Sm, and let B be a subgroup of Sn. We construct new
groups as described below.
Definition 1.1.3. We define two groups A > B and A × B both isomorphic to the
product of A and B, where A > B is a subgroup of Sm+n and A × B is a subgroup
of Smn.
The elements of the product groups are of the form (a, b), where a ∈ A and b ∈ B.
The group operation is defined by
(a1, b1) · (a2, b2) = (a1a2, b1b2),
where a1 and a2 are elements of A, and b1 and b2 are elements of B.
The group A > B acts on the set [m + n] by
(1.1.1) (a, b)(i) =
a(i), if i ∈ 1, 2, . . . , m,
b(i − m) + m, if i ∈ m + 1, m + 2, . . . , m + n.
2
CHAPTER 1. GROUP ACTIONS
Figure 1.1 illustrates the action of an element (a, b) of the group A > B.
b
a
Figure 1.1. Group action of A > B.
The group A × B acts on the set [m] × [n] and may therefore be viewed as a
subgroup of Smn. The action of an element of A × B on an element of [m] × [n] is
given by
(a, b)(i, j) = (a(i), b(j)),
for all i ∈ [m] and j ∈ [n].
Figure 1.2 illustrates the action of an element (a, b) of the group A × B.
b
a
Figure 1.2. Group action of A × B.
Definition 1.1.4. The wreath product of A and B, denoted B ≀ A is the group in
which the elements are ordered pairs (α, τ), where α is a permutation in A and τ is
3
CHAPTER 1. GROUP ACTIONS
a function from [m] to B. An element (α, τ) of B ≀ A acts on the set [m] × [n] by
(α, τ)(i, j) = (αi, τ(i)j),
for all i ∈ [m] and j ∈ [n].
Figure 1.3 illustrates the action of an element (α, τ) of the group B ≀ A.
α
τ(1)
τ(m)
τ(2)
Figure 1.3. Group action of B ≀ A.
The composition of two elements (α, τ) and (β, η) of B ≀ A is given by
(α, τ)(β, η) = (αβ, (τ β)η),
where β ∈ A is viewed as a function from [m] to [m], and (τ β)η denotes the
point-wise multiplication of τ β and η, both functions from [m] to B.
Again B ≀ A can be identified with a subgroup of Smn. Note that the order of
B ≀ A is∣∣B ≀ A
∣∣ = |A| · |B|m.
Example 1.1.5. Let A be the symmetric group of order 2, and let B be the cyclic
group of order 3. The element (α, τ) in B ≀ A, where α = (1, 2), τ(1) = id and
τ(2) = (1, 2, 3), acts on the set [2] × [3] by
(1, 1) 7→ (2, 1) (2, 1) 7→ (1, 2)
(1, 2) 7→ (2, 2) (2, 2) 7→ (1, 3)
(1, 3) 7→ (2, 3) (2, 3) 7→ (1, 1)
4
CHAPTER 1. GROUP ACTIONS
Therefore, the element (α, τ) of B ≀ A is a 6-cycle.
The following Theorem is a generalization of the Cauchy-Frobenius Theorem,
alias Burnside’s Lemma. For the proof of a more general result, with applications
and further references, see Robinson [25]. Another application is given in [6].
Theorem 1.1.6. (Cauchy-Frobenius) Suppose that a finite group M ×N acts on
a set S. The groups M and N , considered as subgroups of M ×N , also act on S. The
group N acts on the set of M-orbits. Then for any g ∈ N , the number of M-orbits
fixed by g is given by
1
|M |
∑
f∈M
fix(f, g),
where fix(f, g) denotes the number of elements in S that are fixed by (f, g) ∈ M ×N .
1.2. Polya’s Cycle Index Polynomials
Definition 1.2.1. Let λ be a partition of n. The power sum symmetric function in
the variables x1, x2, . . . [27, p. 297] indexed by λ, denoted pλ, is defined by
pn = pn[x] =∑
i
xni , n ≥ 1
pλ = pλ[x] = pλ1pλ2 · · · =∏
k≥1
pck(λ)k , if λ = (λ1, λ2, . . . ) = (1c1(λ), 2c2(λ), . . . ).
The pλ form a basis for the ring of symmetric functions in the variables x1, x2, . . . .
We can also define power sum symmetric functions in the variables y1, y2, y3, . . . ,
written as pλ[y], in a similar fashion.
Definition 1.2.2. The operation called plethysm on the ring of symmetric functions
(see Stanley [27, p. 447]) is defined to be such that if f is a symmetric function in
the variables x1, x2, . . . , then the plethysm f pn is obtained from the expression for
5
CHAPTER 1. GROUP ACTIONS
f in terms of the xi’s by replacing each xi with xni . More generally, if f and g are
arbitrary symmetric functions, where g is expressed as a function of the power sum
symmetric functions, i.e., g = g(p1, p2, . . . ), then f g is the polynomial obtained
from f by replacing each variable pk by g(pk, p2k, . . . ).
It follows immediately that pm pn = pmn for all positive integers m, n.
Next, we introduce an operation ⊠ on power sum symmetric functions that was
studied by Harary [10], and by Maia and Mendez [18].
Definition 1.2.3. We define an operation ⊠ on the symmetric functions by letting
pν := pλ ⊠ pµ,
where
ck(ν) =∑
lcm(i,j)=k
gcd(i, j) ci(λ)cj(µ),
in which lcm(i, j) denotes the least common multiple of i and j, and gcd(i, j) de-
notes the greatest common divisor of i and j, and then extending it to all symmetric
functions through bilinearity.
Proposition 1.2.4. The operation ⊠ as defined by Definition 1.2.3 is commuta-
tive and associative. That is, for any partitions λ, µ and ν,
(commutativity) pλ ⊠ pµ = pµ ⊠ pλ,
(associativity) (pλ ⊠ pµ) ⊠ pν = pλ ⊠ (pµ ⊠ pν).
Proof. The commutativity is clear.
For the associativity, we write
(pλ ⊠ pµ) ⊠ pν = pα,
6
CHAPTER 1. GROUP ACTIONS
pλ ⊠ (pµ ⊠ pν) = pβ.
Using the fact that for any positive integers i and j,
ij = lcm(i, j) gcd(i, j),
we calculate the number of cycles of length l, for any integer l, in the partition α as
follows:
cl(α) =∑
lcm(i,j,k)=l
gcd(i, j) gcd(lcm(i, j), k) ci(λ)cj(µ)ck(ν)
=∑
lcm(i,j,k)=l
ij
lcm(i, j)
lcm(i, j) k
lcm(lcm(i, j), k)ci(λ)cj(µ)ck(ν)
=∑
lcm(i,j,k)=l
ijk
lcm(i, j, k)ci(λ)cj(µ)ck(ν)
=∑
lcm(i,j,k)=l
ijk
lci(λ)cj(µ)ck(ν).
The calculation for the partition β is similar.
The properties of ⊠ allow us to talk about the ⊠ operation on a set of partitions.
Let λ(i), i = 1, 2, . . . , l, be partitions of n1, n2, . . . , nl, respectively. We have
pλ(1) ⊠ pλ(2) ⊠ · · · ⊠ pλ(l) = pν ,
where ν is a partition of N = n1n2 · · ·nl. It is easy to verify that
ck(ν) =∑
lcm(j1,j2,...,jl)=k
∏lr=1 jr
k
l∏
i=1
cji
(λ(i)
).
7
CHAPTER 1. GROUP ACTIONS
Definition 1.2.5. Let σ ∈ Sn. The cycle type monomial of σ, denoted Z(σ), is the
monomial in the variables p1, p2, . . . , pn defined by
Z(σ) =
n∏
k=1
pck(σ)k = pc.t.(σ).
In other words, the cycle type monomial of σ is the power sum symmetric function
index by the cycle type of σ.
Let A be a subgroup of Sn. The cycle index polynomial of A, defined by Polya [22,
pp. 64–65], is
Z(A) = Z(A; p1, p2, . . . , pn) =1
|A|
∑
σ∈A
Z(σ).
Polya [21] (translated in [22]) gave the cycle index polynomials of the product
and the wreath product of two permutation groups.
Theorem 1.2.6. (Polya) Let A and B be permutation groups on [m] and [n],
respectively. Let A > B be the product of A and B acting on [m + n], and let B ≀A be
the wreath product of A and B acting on [mn]. Then the cycle index polynomials of
A > B and B ≀ A are
Z(A > B) = Z(A)Z(B),
Z(B ≀ A) = Z(A) Z(B),
where Z(A) Z(B) denotes the plethysm of Z(A) with Z(B).
1.3. Exponentiation Group
Let A be a subgroup of Sm, and let B be a subgroup of Sn. We introduce in the
following another representation of the wreath product (Definition 1.1.4).
8
CHAPTER 1. GROUP ACTIONS
Definition 1.3.1. The exponentiation group BA, studied by Palmer [19], is isomor-
phic to the wreath product B ≀A. That means, the set of elements of BA is the same
as the set of elements of B ≀A, and hence the order of BA is the same as the order of
B ≀ A, i.e.,
|BA| = |A| · |B|m.
However the exponentiation group BA acts on [n]m, identified with the set of functions
from [m] to [n], instead of [m] × [n]. Given any α ∈ A and τ ∈ Bm, the element
(α, τ) ∈ BA sends each element f ∈ [n]m to the element g ∈ [n]m defined by
((α, τ)f)(i) = g(i) = τ(i)(f(α−1i)),
for any i ∈ [m]. Therefor, BA can be identified with a subgroup of Snm.
Figure 1.4 illustrates the action of an element (α, τ) of the group BA.
τ(1)
τ(5
) τ(2)
τ(3)τ(4)
α
α
αα
α
Figure 1.4. Group action of BA.
Example 1.3.2. Let A, B, α and τ be the same as in Example 1.1.5.
9
CHAPTER 1. GROUP ACTIONS
Let f : [2] → [3] be such that f(1) = 1 and f(2) = 3. We see that the image of f
under the action of (α, τ) ∈ BA is the function g : [2] → [3] such that g(1) = 3 and
g(2) = 2, as shown in Figure 1.5.
f(2) = 3
f g(α, τ)
g(1) = 3
g(2) = 2
f(1) = 1
Figure 1.5. The element (α, τ) of BA sends f to g.
The action of (α, τ) on the set of functions from [2] to [3] is illustrated in Figure 1.6.
Hence the cycle type of (α, τ) is (6, 3).
Figure 1.6. The cycle type of (α, τ) is (6, 3).
The cycle index polynomial of the exponentiation group was given by Palmer and
Robinson [20]. They defined the following operators Ik for positive integers k.
Let R = Q [p1, p2, . . . ] be the ring of polynomials with the operation ⊠ as defined
by Definition 1.2.3. Palmer and Robinson defined for positive integers k the Q-linear
operators Ik on R as follows:
10
CHAPTER 1. GROUP ACTIONS
Let λ = (λ1, λ2, . . . ) be a partition of n. The action of Ik on the monomial pλ is
given by
(1.3.1) Ik(pλ) = pγ ,
where γ = (γ1, γ2, . . . ) is the partition of nk with
cj(γ) =1
j
∑
l|j
µ
(j
l
)( ∑
i | l/ gcd(k,l)
ici(λ)
)gcd(k,l)
.
Furthermore, Ik generates a Q-algebra Ω of Q-linear operators on R. For any
elements I, J ∈ Ω, any r ∈ R and a ∈ Q, we set
(aI)(r) = a(I(r)),
(I + J)(r) = I(r) + J(r),
(IJ)(r) = I(r) ⊠ J(r).(1.3.2)
Remark 1.3.3. As discussed in Palmer and Robinson’s proof of Theorem 1.3.6, if
Im(pµ) = pν , then ν is the cycle type of an element (α, τ) of the exponentiation group
BA acting on [n]m, where α is a permutation in A with a single m-cycle, and τ ∈ Bm
is such that
c. t. (τ(m)τ(m − 1) · · · τ(2)τ(1)) = µ.
Example 1.3.4. We take the same A, B, α and τ from Examples 1.1.5. That is,
A = S2, B = C3, α = (1, 2) and τ(1) = id, τ(2) = (1, 2, 3). Then the cycle type of α
is λ = (2), and the cycle type of τ(2)τ(1) is µ = (3). The cycle type of (α, τ) acting
on the set [3]2 is ν = (6, 3), as shown in Figure 1.6, hence
I2(p3) = p3p6.
11
CHAPTER 1. GROUP ACTIONS
Definition 1.3.5. Let f1 and f2 be elements of the ring R = Q [p1, p2, . . . ]. We
define the exponential composition of f1 and f2, denoted f1 ∗ f2, to be the image of
f2 under the operator obtained by substituting the operator Ir for the variables pr in
f1.
Note that the operation ∗ is linear in the left parameters, but not on the right
parameters. We call this the partial linearity of the operation ∗.
Let A be a subgroup of Sm, and let B be a subgroup of Sn. Palmer and Robinson
proved the following formula [20, pp. 128–131] for computing the cycle index poly-
nomial of the exponentiation group BA in terms of the cycle index polynomials of A
and B.
Theorem 1.3.6. (Palmer and Robinson) Let A and B be as described in above.
Then the cycle index polynomial of BA is the exponential composition of Z(A) with
Z(B). That is,
(1.3.3) Z(BA) = Z(A) ∗ Z(B).
Example 1.3.7. Continuing Example 1.3.4, we apply Theorem 1.3.6 to compute
the cycle index of the group BA, where A = S2 and B = C3. Note that the order of
BA is∣∣BA
∣∣ = |A| · |B|2 = 18.
We get the cycle index polynomial of BA
Z(BA) = Z(A) ∗ Z(B)
=
[1
2(p2
1 + p2)
]∗
[1
3(p3
1 + 2p3)
]
12
CHAPTER 1. GROUP ACTIONS
=1
2I1
[1
3(p3
1 + 2p3)
]⊠ I1
[1
3(p3
1 + 2p3)
]+
1
2I2
[1
3(p3
1 + 2p3)
]
=1
2
[1
3I1(p
31) +
2
3I1(p3)
]⊠
[1
3I1(p3
1) +2
3I1(p3)
]+
1
6I2(p3
1) +1
3I2(p3)
=1
18I1(p3
1) ⊠ I1(p31) +
2
9I1(p3) ⊠ I1(p3
1) +2
9I1(p3) ⊠ I1(p3) +
1
6I2(p3
1) +1
3I2(p3)
=1
18p9
1 +2
9p3
3 +2
9p3
3 +1
6p3
1p32 +
1
3p3p6
=1
18(p9
1 + 8p33 + 3p3p
32 + 6p3p6).
The cycle index polynomial Z(BA) tells the structure of the group BA. For
example, the coefficient 6 for the term p3p6 tells that there are 6 elements in BA with
cycle type (6, 3). These elements are precisely those (α, τ), where α is the 2-cycle in
group A, and τ : [2] → B is such that τ(1)τ(2) is a 3-cycle in group B. Moreover, we
observe that the term p33 in Z(BA) with coefficient 8 is contributed in two ways. One
is from those (α, τ) such that α is the identity element of A, and τ(1) and τ(2) are
3-cycles in B, each having two choices; the other is from those (α, τ) such that α is
the identity element of A, one of τ(1) and τ(2) is the identity of B, and the other is
a 3-cycle in B, which gives rise to 4 possibilities.
Remark 1.3.8. Let A and B be the same as in Theorem 1.3.6. Let (α, τ) be an
element of BA, where
c. t.(α) = λ = (λ1, λ2, . . . ),
and τ ∈ Bm is such that the cycle type of τ restricted on each λi-cycle of α is µ(i),
i.e., if the i-th cycle of α is written as
(a1, a2, . . . , aλi),
13
CHAPTER 1. GROUP ACTIONS
then µ(i) is the cycle type of the permutation
τ(a1)τ(a2) · · · τ(aλi).
Then the cycle type of (α, τ), as an element of the group BA acting on the set of
functions from [m] to [n], is ν for which
pν = Iλ1(pµ(1)) ⊠ Iλ2(pµ(2)) ⊠ · · · .
In such cases, for simplicity and without ambiguity, we write
pν = I(λ; µ(1), µ(2), . . . ).
14
CHAPTER 2
Combinatorial Theory of Species
2.1. Definition of Species
The combinatorial theory of species was initiated by Joyal [13, 12]. In short,
species are classes of “labeled structures”. A formal definition (see Bergeron, Labelle,
and Leroux [2, pp. 1–11]) is given in the following.
Definition 2.1.1. Let B be the category of finite sets with bijections. A species (of
structures) is a functor from B to itself, i.e.,
F : B → B.
Given a species F , we obtain for each finite set U a finite set F [U ], which is called
the set of F -structures on U , and for each bijection σ : U → V a bijection
F [σ] : F [U ] → F [V ],
which is called the transport of F -structures along σ.
We denote by F [n] = F [1, 2, . . . , n] the set of F -structures on [n]. The symmet-
ric group Sn acts on the set F [n] by transport of structures. The Sn-orbits under
this action are called unlabeled F -structures of order n.
Example 2.1.2. The species of graphs is denoted by G . Note that by graphs we
mean simple graphs, that is, graphs without loops or multiple edges. Thus defined,
15
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
we mean by G [U ] the set of graphs with vertex set U , and by a G -structure on U a
graph with vertex set U . More formally, the species of structures G generates
• for any finite set U , a set of graphs with vertex set U ;
• for any bijection σ : U → V , a bijection G [σ] : G [U ] → G [V ].
1 53
42
a ec
db
U = 1, 2, 3, 4, 5
V = a, b, c, d, e
σ
Figure 2.1. A bijection σ from U to V induces a bijection G [σ], thetransport of G -structures along σ, sending a graph with vertex set Uto a graph with vertex set V .
Example 2.1.3. We give a list of examples of species.
• The empty species 0 is defined by setting 0[U ] = ∅ for all U .
• The singleton set species 1 is defined by setting
1[U ] =
U, if U = ∅,
∅, otherwise.
• The species of singletons X is defined by setting
X[U ] =
U, if |U | = 1,
∅, otherwise.
• The species of sets E is defined by setting E [U ] = U for each finite set U .
16
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
• The species of linear orders L . In particular, the species of linear orders on
n-element sets is denoted by Xn.
• The species of connected graphs G c.
• The species of complete graphs K .
• The species of permutations Φ.
Example 2.1.4. For any graph G, we define the species associated to G, denoted
OG, to be such that
a) for each finite set U , OG[U ] is the set of graphs isomorphic to G with vertex
set U . Note that L(G) = |OG[n]| .
b) for each bijection σ : U → V , where |U | = |V |, OG[σ] is a bijection from
OG[U ] to OG[V ], sending a graph H isomorphic to G with vertex set U to a
graph OG[σ](H) whose vertex labeling is obtained from the vertex labeling
of H by replacing each label u ∈ U with σ(u).
It is straightforward to see that the bijections OG[σ] further satisfy
i. for all bijections σ : U → V and τ : V → W , OG[τσ] = OG[τ ]OG[σ].
ii. if U = V and σ is the identity map on U , then OG[σ] is also the identity map
on OG[U ].
G OG[a, b, c, d, e]
e
a
b
b e
dc
d a
b c
d
a c
ba
ed
c
b
ce
a
e
d
Figure 2.2. The 5 OG-structures on the vertex set a, b, c, d, e fora given graph G.
17
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Definition 2.1.5. Each species F is associated with three generating series. First,
the exponential generating series of the species F , given by
F (x) =∑
n≥0
|F [n]|xn
n!,
counts labeled F -structures.
Second, the type generating series of the species F , given by
F (x) =∑
n≥0
fn xn,
where fn is the number of unlabeled F -structures of order n, counts unlabeled F -
structures.
The last, but also the most important, associated series is called the cycle index
of the species F :
ZF = ZF (p1, p2, . . . ) =∑
n≥0
(∑
λ⊢n
fix F [λ]pλ
zλ
).
In this definition, fix F [λ] denotes the number of F -structures on [n] fixed by F [σ],
where σ is a permutation of [n] with cycle type λ, and pλ is the power sum symmetric
function indexed by the partitions λ of n.
The following theorem (see [2]) illustrates the importance of the cycle index in
the theory of species.
Theorem 2.1.6. (Bergeron, Labelle and Leroux) For any species of structures F ,
we have
F (x) = ZF (x, 0, 0, . . . ),
F (x) = ZF (x, x2, x3, . . . ).
18
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Remark 2.1.7. Consider the functorial aspects of the species of sets E and the
species of complete graphs K . We see that there is a natural transformation α that
produces for every finite set U a bijection between E [U ] and K [U ], namely, sending
the set U to the complete graph with vertex set U . Furthermore, the following
diagram commutes for any finite sets U , V and any bijection σ : U → V :
E [U ]E [σ]
−−−→ E [V ]
α
yyα
K [U ]K [σ]−−−→ K [V ]
In this case we call these two species isomorphic to each other, denoted E = K .
The general definition of two species being isomorphic to each other is similar. As
pointed out on [2, p. 21], the concept of isomorphism is compatible with the transition
to series. Therefore, we do not make distinctions between isomorphic species during
calculations.
Example 2.1.8. For Φ and L the species of permutations and the species of linear
orders, we notice that their exponential generating series are identical:
Φ(x) = L (x) =1
1 − x,
while their type generating series and cycle indices differ:
Φ(x) =∏
k≥1
1
1 − xk, L (x) =
1
1 − x,
ZΦ =∏
k≥1
1
1 − pk, ZL =
1
1 − p1.
This is because that although there are same number of permutations and linear
orders on a given finite set U , the permutations and linear orders are not transported
19
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
in the same manner along bijections. In fact, a linear order admits only a single
automorphism, while a permutation admits many automorphisms in general.
2 1
342 5
3 5 4 1
a) b)
Figure 2.3. a) A linear order on [5] admits only a single automor-phism. b) A permutation on [5] admits six automorphisms.
Example 2.1.9. Let G be the species of graphs. It is well-known that the exponen-
tial generating series of G is given by
G (x) =∑
n≥0
2(n2)
xn
n!.
The cycle index of G was given on [2, p. 76]:
(2.1.1) ZG =∑
n≥0
(∑
λ⊢n
fix G [λ]pλ
zλ
),
where
fix G [λ] = 212
∑i,j≥1 gcd(i, j) ci(λ)cj(λ)− 1
2
∑k≥1(k mod 2) ck(λ),
in which ci(λ) denotes the number of parts of length i in λ.
The above formulas permit us to write out the first several terms of the associated
series of G using Maple:
G (x) = 1 +x
1!+ 2
x2
2!+ 8
x3
3!+ 64
x4
4!+ 1024
x5
5!+ 32768
x6
6!+ 2097152
x7
7!
+ 68435456x8
8!+ 68719476736
x9
9!+ · · · ,
G (x) = 1 + x + 2x2 + 4x3 + 11x4 + 34x5 + 156x6 + 1044x7 + 12346x8 + · · · ,
20
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
ZG = 1 + p1 + (p21 + p2) +
(4
3p3
1 + 2p1p2 +2
3p3
)
+
(8
3p4
1 + 4p21p2 + 2p2
2 +4
3p1p3 + p4
)
+
(128
15p5
1 +32
3p3
1p2 + 8p1p22 +
8
3p2
1p3 +4
3p2p3 + 2p1p4 +
4
5p5
)+ · · · ,
2.2. Species Operations
We describe in this section some frequently used operations on species, namely,
the sum, product, and substitution of species. Readers are referred to [2, pp. 1–58]
for more detailed definitions of F1 + F2, F1F2, F1(F2), F1 × F2, F •1 , F ′
1 for arbitrary
species F1 and F2.
Definition 2.2.1. The sum of F1 and F2 is denoted by F1+F2. An F1+F2-structure
on a finite set U is either an F1-structure on U or an F2-structure on U .
or
F1 F2F1 + F2
=
Figure 2.4. Addition of species F1 + F2.
The formulae for F1 + F2 are given by
(F1 + F2)(x) = F1(x) + F2(x),
(F1 + F2)(x) = F1(x) + F2(x),
ZF1+F2 = ZF1 + ZF2.
21
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Each species F gives rise to a canonical decomposition
F = F0 + F1 + F2 + · · · + Fn + · · · ,
where Fnn≥0 is the family of species defined by setting, for each n,
Fn[U ] =
F [U ], if |U | = n,
∅, otherwise.
If F = Fn, then we say the species F is concentrated on the cardinality n. We denote
by F+ the species of nonempty F -structures:
F+ = F1 + F2 + · · · + Fn + · · · ,
F+[U ] =
F [U ], if |U | ≥ 1,
∅, if |U | = 0.
Example 2.2.2. The species of sets E and the species of linear orders L can be
decomposed as
E = E0 + E1 + E2 + · · · = 1 + X + E2 + · · · .
L = L0 + L1 + L2 + · · · = 1 + X + X2 + · · · .
Definition 2.2.3. The product of F1 and F2 is denoted by F1 ·F2. An F1F2-structure
on a finite set U is of the form (π; f1, f2), where π is an ordered partition of U with
two blocks U1 and U2, Ui possibly empty, and fi is an Fi-structure on Ui for each i.
The formulae for the associated series of F1F2 are given by
(F1F2)(x) = F1(x)F2(x),
(F1F2)(x) = F1(x)F2(x),
22
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
ZF1F2 = ZF1ZF2.
F1 · F2
F1 F2
=
Figure 2.5. Multiplication of species F1 · F2.
Definition 2.2.4. The composition of F1 and F2 is denoted by F1 F2, or equiva-
lently, F1(F2). An F1(F2)-structure on a finite set U is a tuple of the form (π, f, γ),
where π is a partition of U , f is an F1-structure on π, and γ is a set of F2-structures
on the blocks of π.
==
F1F1 F2
F2
F2
F2
F2
F2
F2F1
Figure 2.6. Composition of species F1 F2.
The formulae for the associated series of F1(F2) are given by
(F1(F2))(x) = F1(F2(x)),
(F1(F2))(x) = ZF1(F2(x), F2(x2), . . . ),
23
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
ZF1(F2) = ZF1 ZF2,
where is the operation of plethysm on symmetric functions defined by Defini-
tion 1.2.2.
Definition 2.2.5. A group A is said to act naturally on a species F if, for any finite
set U , there is an A-action
ρU : A × F [U ] → F [U ]
so that for each bijection σ : U → V , the following diagram commutes:
A × F [U ]ρU−−−→ F [U ]
idA ×F [σ]
yyF [σ]
A × F [V ]ρV−−−→ F [V ]
Definition 2.2.6. The quotient species (see Bergeron, Labelle, and Leroux [2, p. 159])
of F by A, denoted F/A, is defined based on a group A acting naturally on a species
F such that
a) for each finite set U , the set of F/A-structures on U is the set of A-orbits of
F -structures on U , i.e.,
(F/A)[U ] = F [U ]/A,
b) for each bijection σ : U → V , the transport of structures
(F/A)[σ] : F [U ]/A → F [V ]/A
is induced from the bijection F [σ] that sends each orbit of the action of A on F [U ]
to an orbit of the action of A on F [V ].
24
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
The bijections (F/A)[σ] satisfy the functorial properties
(F/A)[τσ] = (F/A)[τ ](F/A)[σ], (F/A)[idU ] = id(F/A)[U ],
because of the functoriality of the species F .
The notion of quotient species appeared in [8] and [3] as an important tool in
combinatorial enumeration.
Example 2.2.7. Let E+ be the species of nonempty sets. Then an (E+ ·E+)-structure
on a finite set U is an ordered pair of nonempty subsets (U1, U2) whose union equals
U . The symmetric group S2 acts naturally on the subscripts of Ui for i = 1, 2, with
orbits unordered pairs of such (U1, U2). Thus we get a quotient species (E+ · E+)/S2,
the species of partitions into two blocks, denoted Ψ(2).
Proposition 2.2.8. Let F1 and F2 be two species. Let A be a group acting natu-
rally on both F1 and F2. Then
(2.2.1) (F1 + F2)/A = F1/A + F2/A.
Proof. An F1 + F2-structure on a finite set U is either an F1-structure on U or
an F2-structure on U . It follows immediately that an orbit of A acting on the set of
F1 + F2-structures on U is either an orbit of A acting on the set F1[U ] or an orbit of
A acting on the set F2[U ].
2.3. Molecular Species
Definition 2.3.1. A species of structures M is said to be molecular [31, 30] if there
is only one isomorphism class of M-structures, i.e., if two arbitrary M-structures are
isomorphic.
25
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
In other words, the species M is molecular if and only if M 6= 0, M is concentrated
on n, and any element in the set M [n] can be sent through some transport of structures
to any other element in M [n].
We introduce in the following a construction of, for any subgroup A of Sn, a
molecular species Xn/A (see Bergeron, Labelle, and Leroux [2, p. 144]). These species
satisfy that Xn/A = Xn/B if and only if A and B are conjugates in Sn. In fact, if M
is any molecular species concentrated on the cardinality n, and s is any M-structure,
then M is the same as the species Xn/A, as defined in below, for some A that is the
automorphism group of s.
Definition 2.3.2. Let n be a non-negative integer, A a subgroup of Sn, U and V
finite sets of cardinality n, and τ : U → V a bijection. Define a species Xn/A to be
such that
a) the set of Xn/A-structures on U is the set of (left) cosets with respect to A of
bijections [n] → U . That is,
(Xn/A)[U ] = σA | σ : [n] → U,
where σA = σ a | a ∈ A.
b) the transport
(Xn/A)[τ ] : (Xn/A)[U ] → (Xn/A)[V ]
is defined by setting
(Xn/A)[τ ](σA) = (τσ)A.
Example 2.3.3. a) There is only one element in the set En[n], so that the condition
“any two elements in En[n] are isomorphic” is automatically true. Hence the species
26
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
En is a molecular species, and
En =Xn
Sn.
b) We denote by Cn the species of oriented cycles of length n. We see that Cn is
a molecular species, and
Cn =Xn
Cn,
where Cn is the cyclic group of order n, i.e., Cn is the subgroup of Sn in which all
elements are n-cycles.
Remark 2.3.4. Let U be a finite set of cardinality n, and let A be a subgroup of
Sn. Then A acts naturally on the set of linear orders on U . If we call the orbit of a
linear order s on U under this A-action the A-orbit of s, then the Xn/A-structures
on U is just the set of A-orbits of the action of A on the set of linear orders on U ,
which is also, as pointed out on [2, p. 160], the quotient species of Xn by A.
The following proposition (see [15, p. 117] Example 7.4) illustrates the close con-
nection between Polya’s cycle index polynomial and the cycle index of a species.
Proposition 2.3.5. Let A be a subgroup of Sn. Then
Z(A) = ZXn/A.
Proof. Recall that the set (Xn/A)[n] is the set of cosets of A in Sn. We set
(Xn/A)[n] = g1A, g2A, . . . , gkA,
where g1, g2, . . . , gk are coset representatives. Then according to the definition of cycle
index series, we have
ZXn/A =1
n!
∑
τ∈Sn
fixXn
A[τ ] Z(τ),
27
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
where Z(τ) is the cycle type polynomial of τ , and the number of elements in (Xn/A)[n]
fixed by τ is
fixXn
A[τ ] = |i ∈ [k] : τgiA = giA|
=∣∣i ∈ [k] : g−1
i τgi ∈ A∣∣ .
We notice that if for some i, g−1i τgi ∈ A, then every element g in the coset represented
by giA satisfies g−1τg ∈ A. We notice also that each coset giA contains exactly |A|
elements.
Hence we get
fixXn
A[τ ] =
1
|A|
∣∣g ∈ Sn : g−1τg ∈ A∣∣ .
Therefore,
ZXn/A =1
n!
1
|A|
∑
τ∈Sn
∣∣g ∈ Sn : g−1τg ∈ A∣∣ Z(τ)
=1
n!
1
|A|
∑
τ, g∈Sng−1τg∈A
Z(τ) =1
|A|
∑
g∈Snσ∈A
1
n!Z(σ) =
1
|A|
∑
σ∈A
Z(σ) = Z(A).
Each subgroup A of Sn corresponds to a molecular species concentrated on the
cardinality n, namely, Xn/A. Proposition 2.3.5 says that Polya’s cycle index poly-
nomial of A is the same as the cycle index of the species Xn/A, hence these two
definitions are equivalent in the case of molecular species.
Example 2.3.6. There are dλ = n!/zλ permutations with cycle type λ in Sn. Thus
the cycle index polynomial of Sn, which is the same as the cycle index of the species
28
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
En, is
ZEn= Z(Sn) =
1
n!
∑
λ⊢n
dλpλ =∑
λ⊢n
pλ
zλ.
Example 2.3.7. Recall for any graph G, OG is the species associated to G. We
oberve that OG is indeed the molecular species Xn/ aut(G), where n is the num-
ber of vertices in G, and aut(G) is the automorphism group of G. It follows from
Proposition 2.3.5 that the cycle index of OG is the cycle index polynomial of the
automorphism group of G. That is,
ZOG= Z(aut(G)).
For example, the cycle index of the graph G in Figure 2.2 is
ZOG= Z(A),
where A is the automorphism group of G, which is the subgroup of S5 isomorphic to
S4. To be more precise,
ZOG=
1
24(p5
1 + 6p31p2 + 8p2
1p3 + 3p1p22 + 6p1p4).
We see from Definition 2.3.1 that molecular species are indecomposable under
addition. This observation leads to a molecular decomposition of any species [2,
p. 141]:
Proposition 2.3.8. Every species of structures F is the sum of its molecular
subspecies:
F =∑
M⊆FM molecular
M.
29
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Example 2.3.9. The molecular decomposition of the species of graphs G is given
on [2, p. 421]:
G = 1+X +2E2 +(2E3 +2XE2)+(2E4 +2E2E2 +2XE3 +2E22 +2X2
E2 +E X2)+ · · ·
Consider the molecular species Xm/A and Xn/B. That means A is subgroup
of Sm and B is a subgroup of Sn. Yeh [31, 30] proved the following theorem for
molecular species.
Theorem 2.3.10. (Yeh) Let A be a subgroup of Sm, and B a subgroup of Sn with
n ≥ 1. We have
Xm
A·Xn
B=
Xm+n
A > B,
Xm
A
Xn
B=
Xmn
B ≀ A,
where the product group A > B acts on the set [m + n], and the wreath product group
B ≀ A, as defined by Definition 1.1.4, acts on the set [mn].
Note that Yeh’s results agree with Polya’s Theorem 1.2.6 for the cycle index
polynomials of A > B and B ≀ A.
In fact, as mentioned in Remark 2.3.4, we think of an Xm/A-structure on the set
U1 with |U1| = m as an A-orbit of linear orders on U1, and an Xn/B-structure on the
set U2 with |U2| = n as a B-orbit of linear orders on U2. Then an (Xm/A) · (Xn/B)-
structure on U = U1∨U2, where ∨ denotes the disjoint union admits an automorphism
group isomorphic to the product group A > B, as shown in Figure 2.7.
We also observe that an (Xm/A) (Xn/B)-structure is an Xm/A-structure on a
set of Xn/B-structures, which is the same as a set of B-orbits of linear orders on [n],
30
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
B-orbits
A-orbits
Figure 2.7. (Xm/A) · (Xn/B) = Xm+n/(A × B).
as shown in Figure 2.8, and hence admits an automorphism group isomorphic to the
wreath product B ≀ A.
B-orbits
A-orbits
B-orbits
B-orbits
Figure 2.8. (Xm/A) (Xn/B) = Xmn/(B ≀ A).
2.4. Compositional Inverse of E+
The species 1 + X is the sum of the singleton set species 1 and the species of
singletons X. A 1 + X-structure on [n] is
(1 + X)[n] =
∅, if n = 0,
1, if n = 1,
∅, otherwise.
31
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
The associated series of 1 + X are
(1 + X)(x) = 1 + x,
1 + X(x) = 1 + x,
Z(1+X) = 1 + p1.
A virtual species is, roughly speaking, the formal difference of species [2, p. 121].
Since the species 1 + X satisfies (1 + X)(0) = 1, Proposition 18 of [2, p. 129] gives
that there exists a unique virtual species Γ, so-called “connected (1 + X)-structures”
as defined on [2, p. 121], with
(2.4.1) 1 + X = E (Γ).
In fact, Equation (2.4.1) leads to
X = E+(Γ),
which implies that Γ is the compositional inverse of E+, written as
Γ = E〈−1〉+ (X).
The associated series of Γ are given by [2, p. 131]:
Γ(x) = log(1 + X)(x) = log(1 + x),
Γ(x) =∑
k≥1
µ(k)
klog ˜(1 + X)(xk) =
∑
k≥1
µ(k)
klog(1 + xk),
ZΓ =∑
k≥1
µ(k)
klog Z(1+X) pk =
∑
k≥1
µ(k)
klog(1 + pk),
32
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
where µ denotes the Mobius function, defined by
µ(k) =
0, if n has one or more repeated prime factors,
1, if n = 1,
(−1)j , if n is a product of j distinct primes.
In fact, we can compute the first several terms of the associated series of Γ:
Γ(x) =x
1!−
x2
2!+ 2
x3
3!− 6
x4
4!+ 24
x5
5!− 120
x6
6!+ 720
x7
7!− . . . ,
Γ(x) = x − x2,
(2.4.2)
ZΓ = p1 −
(1
2p2
1 +1
2p2
)+
(1
3p3
1 −1
3p3
)−
(1
4p4
1 −1
4p2
2
)+
(1
5p5
1 −1
5p5
)+ · · · .
Read [23, p. 4] derived identity (2.4.2) as the type generating series of the com-
positional inverse of the species E+.
Example 2.4.1. Let G c be the species of connected graphs, and E the species of
sets. The observation that every graph is a set of connected graphs gives rise to the
following species identity:
(2.4.3) G = E (G c),
which can be read as “a graph is a set of connected graphs”, and gives rise to the
identity
Gc = Γ(G+),
which leads to the enumeration of connected graphs:
Gc(x) = log(G (x)),
33
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
G c(x) =∑
k≥1
µ(k)
klog(G (xk)),
ZG c =∑
k≥1
µ(k)
klog(ZG pk − pk + 1).
Using Maple, we can compute the first several terms of the associated series of
G c:
Gc (x) =
x
1!+
x2
2!+ 4
x3
3!+ 38
x4
4!+ 728
x5
5!+ 26704
x6
6!+ 1866256
x7
7!(2.4.4)
+ 251548592x8
8!+ 66296291072
x9
9!+ . . . ,(2.4.5)
G c(x) = x + x2 + 2x3 + 6x4 + 21x5 + 112x6 + 853x7 + 11117x8 + · · · ,
ZG c = p1 +
(1
2p2
1 +1
2p2
)+
(1
3p3 +
2
3p3
1 + p1p2
)
+
(19
12p4
1 + 2p21p2 +
5
4p2
2 +2
3p1p3 +
1
2p4
)
+
(193 p3
1p2 +2
3p2p3 +
91
15p5
1 + 5p1p22 +
4
3p2
1p3 +3
5p5 + p1p4
)
+
(1669
45p6
1 +91
3p4
1p2 +38
9p3
1p3 +43
2p2
1p22 + 2p2
1p4 +8
3p1p2p3
+4
5p1p5 +
26
3p3
2 +5
2p2p4 +
25
18p2
3 +5
6p6
)+ · · · .(2.4.6)
Recall Proposition 2.3.8 that states that there is a molecular decomposition of any
species up to isomorphism. We can often identify a given species in terms of molecular
species using combinatorial operators. Figure 2.9 shows unlabeled connected graphs
on no more than 4 vertices.
34
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Figure 2.9. Unlabeled connected graphs on n vertices, n ≤ 4.
In this way, the molecular decomposition of the species of connected graphs G c
takes the form
Gc = X + E2 +
(XE2 + E3
)+
(E2 X2 + XE3 + X2
E2 + E2E2 + E4 + D4
)+ · · · ,
where D4 is the molecular species corresponding to the dihedral group D4 of order 4,
that is, D4 = X4/D4, and
ZD4 = Z(D4) =1
8
(p4
1 + 2p21p2 + 3 p2
2 + 2p4
).
2.5. Multisort Species
The theory of multisort species [2, p. 100] is analogous to multivariate functions.
Definition 2.5.1. A k-set U is a k-tuple of sets
U = (U1, U2, . . . , Uk).
A multifunction f from (U1, U2, . . . , Uk) to (V1, V2, . . . , Vk), denoted
f : (U1, U2, . . . , Uk) → (V1, V2, . . . , Vk),
35
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
is a k-tuple of functions f = (f1, f2, . . . , fk) such that fi : Ui → Vi for i = 1, . . . , k.
Such f is called bijective if each fi is a bijection.
Let Bk be the category of k-sets with bijective multifunctions.
Definition 2.5.2. A species of k sorts, where k ≥ 1, is a functor from the category
of finite k-sets Bk to the category of finite sets B, i.e.,
F : Bk → B.
A k-sort species F gives rise for each k-set
U = (U1, U2, . . . , Uk)
a finite set F [U1, U2, . . . , Uk], and for each bijective multifunction
σ = (σ1, σ2, . . . , σk) : (U1, U2, . . . , Uk) → (V1, V2, . . . , Vk)
a bijection
F [σ] = F [σ1, σ2, . . . , σk] : F [U1, U2, . . . , Uk] → F [V1, V2, . . . , Vk]
that is called the transport of F -structures along σ.
We denote by F (X, Y ) a 2-sort species F . We use the notation [m, n], where m
and n are positive integers, to denote the 2-set (1, 2, 3, . . . , m, 1, 2, 3, . . . , n).
The exponential generating series of F (X, Y ) is
F (x, y) =∑
m,n≥0
∣∣F [m, n]∣∣ xm
m!
yn
n!,
where F [m, n] = F [1, 2, . . . , m, 1, 2, . . . , n] is the number of labeled F -structures
on the 2-set [m, n].
36
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
F
Figure 2.10. Multisort species.
The type generating series of F (X, Y ) is
F (x, y) =∑
m,n≥0
fm,n xmyn,
where fm,n is the number of unlabeled F -structures in which the cardinality of the
first sort set is m and the cardinality of the second sort set is n.
The cycle index of F (X, Y ) is
ZF (X,Y ) =∑
m,n≥0
( ∑
λ⊢m, µ⊢n
fix F [λ, µ]pλ[x]
zλ
pµ[y]
zµ
),
where pλ[x] and pλ[y] are power sum symmetric functions in variable sets x1, x2, . . .
and y1, y2, . . . , respectively, fix F [λ, µ] denotes the number of F -structures on [m, n]
fixed by F [σ, π], where σ is a permutation of [m] with cycle type λ and π is a permu-
tation of [n] with cycle type µ.
Example 2.5.3. Let A (X, Y ) be the 2-sort species of rooted trees whose internal
vertices are of sort X and whose leaves are of sort Y . Figure 2.11 shows the transport
37
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
of structures of two A (X, Y )-structures under a bijective multifunction
σ =
(2
j
3
a
4
g
5
m
7
f
8
e
9
d
12
n
∣∣∣∣1
l
6
k
10
h
11
b
13
i
14
c
):
A [σ]mf
e
ch
g
n
d
k bi
j
a
l
4
12
9
6 1113
2
57
8
1410 1
3
Figure 2.11. The transport of structures A [σ] sends an element ofA [U1, U2] to an element of A [V1, V2], where
U1 = 2, 3, 4, 5, 7, 8, 9, 12, U2 = 1, 6, 10, 11, 13, 14,V1 = j, a, g, m, f, e, d, n, V2 = l, k, h, b, i, c.
2.6. Arithmetic Product of Species
The arithmetic product was studied by Manuel Maia and Miguel Mendez [18].
They developed a decomposition of a set, called a rectangle, which, as we shall see, is
essentially an equivalence class of high dimensional linear orders.
Definition 2.6.1. Let U be a finite set. Suppose |U | = ab. A rectangle on U of
height a is a pair (π1, π2) such that π1 is a partition of U with a blocks, each of size
b, and π2 is a partition of U with b blocks, each of size a, and if B is a block of π1
and B′ is a block of π2 then |B ∩ B′| = 1.
38
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Figure 2.12 shows two representations of a rectangle on [12] of height 3. They
both represent the rectangle (π1, π2).
π1 = 1, 3, 4, 9, 2, 5, 8, 10, 6, 7, 11, 12,
π2 = 1, 8, 12, 3, 5, 6, 2, 4, 11, 7, 9, 10.
4
4
611712
1 9 3
52108
1
28
12 11
3
5
6
9
10
7
∼
Figure 2.12. A rectangle on [12] of height 3.
Equivalently we can represent a rectangle as a bipartite graph. A rectangle which
is the same as in Figure 2.12 is shown by Figure 2.13, where the labels are on the
10
9 8 12
4
3
1
2
5
711
6
π1 = A1, A2, A3
B1
B2
B3
B4
A1
A2
A3 π2 = B1, B2, B3, B4
Figure 2.13. A rectangle on [12] of height 3.
edges, and the vertex sets on the left and on the right represent partitions π1 and π2.
More precisely, each edge x in the bipartite graph corresponds to exactly one pair of
vertices (Ai, Bj) with Ai ∈ π1 and Bj ∈ π2, and this can be interpreted as x being
the only element in the intersection of Ai and Bj .
39
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Definition 2.6.2. Let U be a finite set. A k-rectangle on U is a k-tuple of partitions
(π1, π2, . . . , πk) such that
a) for each i ∈ [k], πi has ai blocks, each of size |U |/ai, where |U | =k∏
i=1
ai.
b) for any k-tuple (B1, B2, . . . , Bk), where Bi is a block of πi for each i ∈ [k],
we have |B1 ∩ B2 ∩ · · · ∩ Bk| = 1.
Figure 2.14 shows a 3-rectangle (π1, π2, π3), represented by a 3-partite graph, and
labeled are on the triangles.
C1
π3 = C1, C2
π1 = A1, A2, A3, A4
A4A3A2A1
C2
B3
B2
B1π2 = B1, B2, B3
Figure 2.14. A 3-rectangle labeled on the triangles.
We denote by N the species of rectangles, and by N (k) the species of k-rectangles.
Let n =∏k
i=1 ai, and let ∆ be the set of bijections of the form
δ : [a1] × [a2] × · · · × [ak] → [n].
Note that the cardinality of the set ∆ is n!. The group
k∏
i=1
Sai= σ = (σ1, σ2, . . . , σk) : σi ∈ Sai
acts on the set ∆ by setting
(σ · δ)(i1, i2, . . . , ik) = δ(σ1(i1), σ2(i2), . . . , σk(ik)),
40
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
for each (i1, i2, . . . , ik) ∈ [a1] × [a2] × · · · × [ak]. We observe that this group action
result in a set of∏k
i=1 Sai-orbits, and that each orbit consists of exactly a1!a2! · · ·ak!
elements of ∆. Observe further that there is a one-to-one correspondence between the
set of∏k
i=1 Sai-orbits on the set ∆ and the set of k-rectangles of the form (π1, . . . , πk),
where each πi has ai blocks. Therefore, the number of such k-rectangles is
n
a1, a2, . . . , ak
:=
n!
a1!a2! · · ·ak!.
Definition 2.6.3. Let F1 and F2 be species of structures with F1[∅] = F2[∅] = ∅.
The arithmetic product of F1 and F2, denoted F1 ⊡ F2, is defined by setting for each
finite set U ,
(F1 ⊡ F2)[U ] =∑
(π1 π2)∈N [U ]
F1[π1] × F2[π2],
where the sum represents the disjoint union. In other words, an F1 ⊡ F2-structure on
a finite set U is a tuple of the form ((π1, f1), (π2, f2)), where fi is an Fi-structure on
the blocks of πi for each i.
A bijection σ : U → V sends a partition π of U to a partition π′ of V , namely,
σ(π) = π′ = σ(B) : B is a block of π. Thus σ induces a bijection σπ : π → π′,
sending each block of π to a block of π′.
The transport of structures for any bijection σ : U → V is defined by
(F1 ⊡ F2)[σ]((π1, f1), (π2, f2)) = ((π′1, F1[σπ1 ](f1)), (π′
2, F2[σπ2 ](f2))).
The arithmetic product on species satisfies the following properties. They are
straightforward to check using the definition.
41
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
F2F1 F2 F1
X
X
X
X
X
X
X
X
X
XX
X
XX
X
Figure 2.15. Arithmetic product F1 ⊡ F2.
Proposition 2.6.4. (Maia and Mendez) Let F1, F2 and F3 be species of structures
with Fi[∅] = ∅ for i = 1, 2, 3, and let X be the species of singleton sets. Then we have
(commutativity) F1 ⊡ F2 = F2 ⊡ F1,
(associativity) F1 ⊡ (F2 ⊡ F3) = (F1 ⊡ F2) ⊡ F3,
(distributivity) F1 ⊡ (F2 + F3) = F1 ⊡ F2 + F1 ⊡ F3,
(unit) F1 ⊡ X = X ⊡ F1 = F1.
FF
X
X
X
X
X
X X
Figure 2.16. X is the unit of the arithmetic product: F ⊡ X = F .
The following proposition about the arithmetic product of two molecular species
could be taken as an alternative definition of the arithmetic product.
Proposition 2.6.5. Let Xm/A and Xn/B be two molecular species. That is, A
is a subgroup of Sm, and B is a subgroup of Sn. Then
Xm
A⊡
Xn
B=
Xmn
A × B,
42
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
where A×B is the product group acting on the set [mn] as defined by Definition 1.1.3.
Example 2.6.6. Recall that the molecular species E2 is the same as the species
X2/S2. The arithmetic product of the species E2 with itself is a species concentrated
on the cardinality 4. To be more precise, the set of E2 ⊡ E2-structures on a set U
with |U | = 4 are obtained by taking the S2 × S2-orbits of 2 by 2 squares filled with
elements of U , where the group S2×S2 acts by switching the rows and the columns.
In other words, E2 ⊡ E2 = X4/(S2 × S2).
E2
a b
dc
E2
Figure 2.17. The species E2 ⊡ E2 is isomorphic to the speciesX4/(S2 × S2).
We can quickly verify Proposition 2.6.5 by observing that the Xm/A-structures
are A-orbits of linear orders on [m], and the Xn/B-structures are B-orbits of linear
orders on [n], and hence an (Xm/A)⊡(Xn/B)-structure on [mn] is just a matrix whose
rows and columns are A-orbits of linear orders on [m] and B-orbits of linear orders on
[n], respectively. The automorphism group of such an (Xm/A) ⊡ (Xn/B)-structure
is therefore isomorphic to the product group A × B.
The associativity of the arithmetic product allows us to talk about the arithmetic
product of a set of species.
43
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
B-orbits
A-orbits
Figure 2.18. (Xm/A) ⊡ (Xn/B) = Xmn/(A × B).
Definition 2.6.7. The arithmetic product of species F1, F2, . . . , Fk with Fi(∅) = ∅
for all i is defined by setting
k
⊡i=1
Fi = F1 ⊡ F2 ⊡ · · ·⊡ Fk,
which sends each finite set U to the set
k
⊡i=1
Fi[U ] =∑
F1[π1] × F2[π2] × · · · × Fk[πk],
where the sum is taken over all k-rectangles (π1, π2, . . . , πk) of U , and represents the
disjoint union. We denote by F ⊡k the arithmetic product of k copies of F .
For each bijection σ : U → V , the transport of structures ofk
⊡i=1
Fi along σ sends
ank
⊡i=1
Fi-structure on U of the form
((π1, f1), (π2, f2), . . . , (πk, fk))
to andk
⊡i=1
Fi-structure on V of the form
((π′1, F1[σπ1 ]f1), (π′
2, F2[σπ2 ]f2), . . . , (π′k, Fk[σπk
]fk)),
44
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
where σπiis the bijection induced by σ sending blocks of πi to blocks of π′
i.
Proposition 2.6.8. (Maia and Mendez) We have the following identities for the
species of k-rectangles.
N = E+ ⊡ E+,
N(k) = E
⊡k+ ,
N(i+j) = N
(i)⊡ N
(j).
Proof. These identities follow straightforwardly from the definitions of rectangle
and arithmetic product of species.
Figures 2.17 and 2.19 show that the set E2⊡E2[4] is the same as the set of rectangles
on [4] of height 2.
E2
1 2
43
E2 E2E2
4
1 3
2
21
4 3E2E2E2
1 3
2 4
E2 E2E2
23
1 4 1 4
32E2E2
Figure 2.19. The E2 ⊡ E2-structures on the set [4].
Maria and Mendez [18] gave the formula for the cycle index of the arithmetic
product.
45
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
Theorem 2.6.9. (Maria and Mendez) Let species F1 and F2 satisfy
F1[∅] = F2[∅] = ∅.
Then we have
(2.6.1) ZF1⊡F2 = ZF1 ⊠ ZF2,
where the operation ⊠ on the power sum symmetric functions on the right-hand side
of the equation is given by Definition 1.2.3.
We notice that due to the linearities of both the arithmetic product of species and
the operation ⊠ on power sum symmetric functions, identity (2.6.1) reduces to the
case when F1 and F2 are both molecular species.
Suppose, say, F1 = Xm/A and F2 = Xn/B for some integers m, n, and some
A and B subgroups of Sm and Sn, respectively. We apply Proposition 2.6.5 and
Proposition 2.3.5 to translate the formula
Z(Xm/A)⊡(Xn/B) = ZXm/A ⊠ ZXn/B
into
(2.6.2) Z(A × B) = Z(A) ⊠ Z(B).
Identity (2.6.2) can be roughly verified by observing that if a is an element of A
and b is an element of B, then the cycle type of (a, b) in the group A × B satisfies
pc.t.((a,b)) = pc.t.(a) ⊠ pc.t.(b).
46
CHAPTER 2. COMBINATORIAL THEORY OF SPECIES
This is because each pair of a k-cycle in a and an l-cycle in b generates cycles of
length lcm(k, l) in the group A × B with multiplicity gcd(k, l), and that the formula
for such pairs of k-cycles in a and l-cycles in b is
pikk ⊠ pjl
l = pgcd(k,l) ikjl
lcm(k,l) .
Figure 2.20 gives an illustration of p3 ⊠ p4 = p12.
Figure 2.20. A 3-cycle in the group A and a 4-cycle in the group Bgenerates a 12-cycle in the group A × B.
Summing up on all different cycle lengths of a and b, we get the desired result.
47
CHAPTER 3
Cartesian Product of Graphs and Prime Graphs
3.0. Introduction
In this chapter, we look at the well-known operation of Cartesian product (Defini-
tion 3.1.1) on graphs, and the notion of prime graphs (Definition 3.1.3) with respect
to Cartesian multiplication. Our goal is to find the cycle index of the species of prime
graphs.
First, we get a relation (Proposition 3.1.8) between the Cartesian product of
graphs and the arithmetic product of species (Definition 2.6.3). We then use the
tool of Dirichlet exponential generating series (Definition 3.2.3) to get a formula
(Theorem 3.2.9) expressing the number of labeled prime graphs in terms of the number
of labeled connected graphs. Furthermore, we see that the unique factorization of a
nontrivial connected graph into the product of powers of prime graphs (Sabidussi [26])
gives a free commutative monoid structure on the set of unlabeled connected graphs,
and leads to a formula (Theorem 3.3.10) that counts unlabeled prime graphs in terms
of unlabeled connected graphs.
In terms of species, the arithmetic product of species studied by Maia and Mendez [18]
comes back into play. As it turns out, we need a stronger notion than arithmetic
product in order to express the species of connected graphs in terms of the species
of prime graphs. We therefore define a new operation, the exponential composition
of species (Definition 3.5.1), which is related to the arithmetic product of species as
48
CHAPTER 3. PRIME GRAPHS
the composition of species is related to the multiplication of species, and get a for-
mula (Theorem 3.6.2) expressing the species of connected graphs as the exponential
composition of the species of prime graphs. An explicit formula for the inverse of
exponential composition would be nice to find, but that problem remains open.
The enumeration of the species of prime graphs is therefore complete by applying
the enumeration theorem (Theorem 3.4.4) for the exponential composition of species,
which is a generalization of the enumeration theorem by Palmer and Robinson about
the cycle index polynomial of the exponentiation group (Theorem 1.3.6).
3.1. Cartesian Product of Graphs
For any graph G, we let V (G) be the vertex set of G, E(G) the edge set of G,
and l(G) = |V (G)| the number of vertices in G. Two graphs G and H with the
same number of vertices are said to be isomorphic, denoted G ∼= H , if there exists a
bijection from V (G) to V (H) that preserves adjacency. Such a bijection is called an
isomorphism from G to H . In the case when G and H are identical, this bijection
is called an automorphism of G. The collection of all automorphisms of G, denoted
aut(G), constitutes a group called the automorphism group of G.
We call the isomorphism classes of graphs unlabeled graphs. If G is a graph with
n vertices, L(G) is the number of graphs isomorphic to G with vertex set [n]. It is
well-known that
L(G) =l(G)!
|aut(G)|.
We use the notation∑n
i=1 Gi = G1 + G2 + · · ·+ Gn to mean the disjoint union of
a set of graphs Gii=1,...,n.
Definition 3.1.1. The Cartesian product of graphs G1 and G2, denoted G1 ⊙ G2,
as defined by Sabidussi [26] under the name the weak Cartesian product, is the graph
49
CHAPTER 3. PRIME GRAPHS
whose vertex set is
V (G1 ⊙ G2) = V (G1) × V (G2) = (u, v) : u ∈ V (G1), v ∈ V (G2),
in which (u, v) is adjacent to (w, z) if
either u = w and v, z ∈ E(G2) or v = z and u, w ∈ E(G1).
For simplicity and without ambiguity, we call G1 ⊙G2 the product of G1 and G2.
2
3,3’
4,3’3,2’
2,2’
2,3’
1,3’
1,2’3,1’
4,1’2,1’
1,1’
1’
2’3’
1
3
4
4,2’
Figure 3.1. The Cartesian product of a graph with vertex set1, 2, 3, 4 and a graph with vertex set 1′, 2′, 3′ is a graph with vertexset (i, j), where i ∈ [4] and j ∈ [3]′.
The following properties of the Cartesian multiplication can be verified straight-
forwardly.
Proposition 3.1.2. Let G1, G2 and G3 be graphs. Then
(commutativity) G1 ⊙ G2∼= G2 ⊙ G1,
(associativity) (G1 ⊙ G2) ⊙ G3∼= G1 ⊙ (G2 ⊙ G3).
50
CHAPTER 3. PRIME GRAPHS
These properties allow us to talk about the Cartesian product of a set of graphs
Gii∈I , denoted ⊙i∈I
Gi. We denote by Gn the Cartesian product of n copies of G.
Definition 3.1.3. A graph G is prime with respect to Cartesian multiplication if G
is a connected graph with more than one vertex such that G ∼= H1 ⊙H2 implies that
either H1 or H2 is a singleton vertex.
Two graphs G and H are called relatively prime with respect to Cartesian mul-
tiplication, if and only if G ∼= G1 ⊙ J and H ∼= H1 ⊙ J imply that J is a singleton
vertex.
We denote by P the species of prime graphs. We also denote by C the set of
unlabeled connected graphs, and by P the set of unlabeled prime graphs.
If G is a connected graph, then G can be decomposed into prime factors, that is,
there is a set Pii∈I of prime graphs such that G ∼= ⊙i∈I
Pi. In Figure 3.2, a connected
graph G with 24 vertices is decomposed into the product of prime graphs P1, P2, P3
with 3, 2 and 4 vertices, respectively.
=
Figure 3.2. The decomposition of a connected graph into prime graphs.
Furthermore, we have the following theorem given by Sabidussi [26].
Theorem 3.1.4. (Sabidussi) For any non-trivial connected graph G, the factor-
ization of G into the Cartesian product of prime powers is unique up to isomorphism.
51
CHAPTER 3. PRIME GRAPHS
The automorphism groups of the Cartesian product of a set of graphs was studied
by Sabidussi [26] and Palmer [19].
Theorem 3.1.5. (Sabidussi) Let Gii=1,...,n be a set of graphs. Then the auto-
morphism group of the Cartesian product of Gii=1,...,n is isomorphic to the auto-
morphism group of the sum of Gii=1,...,n. That is,
aut
(n⊙i=1
Gi
)∼= aut
( n∑
i=1
Gi
).
Theorem 3.1.6. (Sabidussi) Let G1, G2, . . . , Gn be connected graphs which are
pairwise relatively prime with respect to Cartesian multiplication. Then the automor-
phism group of the Cartesian product of Gii=1,...,n is isomorphic to the product of
each of the automorphism groups of Gii=1,...,n. That is,
(3.1.1) aut
(n⊙i=1
Gi
)∼=
n∏
i=1
aut(Gi).
Example 3.1.7. The prime graphs P1, P2 and P3 in Figure 3.2 are pairwise relatively
prime, so the automorphism group of G is
aut(G) = aut(P1) × aut(P2) × aut(P3)
= S2 × S2 × V4,
where V4∼= S2 × S2.
Recall the arithmetic product of species defined by Definition 2.6.3 and the species
associated to a graph defined in Example 2.1.4. We present in the following the close
connection between the arithmetic product and the Cartesian product.
52
CHAPTER 3. PRIME GRAPHS
Proposition 3.1.8. Let G1 and G2 be two graphs that are relatively prime to each
other. Then the species associated to the Cartesian product of G1 and G2 is equivalent
to the arithmetic product of the species associated to G1 and the species associated to
G2. That is,
OG1⊙G2 = OG1 ⊡ OG2
Proof. Let l(G1) = m and l(G2) = n. Then l(G1 ⊙ G2) = mn.
Since G1 and G2 are relatively prime, applying Theorem 3.1.6 we get
aut(G1 ⊙ G2) = aut(G1) × aut(G2).
Therefore,
OG1⊙G2 =X l(G1⊙G2)
aut(G1 ⊙ G2)=
Xmn
aut(G1) × aut(G2)
=Xm
aut(G1)⊡
Xn
aut(G2)= OG1 ⊡ OG2 .
Note that if G1 and G2 are not relatively prime to each other, then the species
associated to the Cartesian product of G1 and G2 is generally different from the
arithmetic product of OG1 and OG2 . This is because that the automorphism group
of the product of the graphs is no longer the product of the automorphism groups of
the graphs. For example, we will see in Remark 3.2.2 that for any prime graph P ,
aut(P ⊙ P ) = aut(P )S2,
and aut(P )S2 is generally much larger than aut(P )2.
53
CHAPTER 3. PRIME GRAPHS
3.2. Labeled Prime Graphs
We introduce in the following an important theorem by Sabidussi about the au-
tomorphism group of a connected graph using its prime factorization.
Theorem 3.2.1. (Sabidussi) Let G be a connected graph with prime factorization
G ∼= P s11 ⊙ P s2
2 ⊙ · · · ⊙ P sk
k ,
where for r = 1, 2, . . . , k, all Pr are distinct prime graphs, and all sr are positive
integers. Then we have
aut(G) ∼=
k∏
r=1
aut(P sr
r ) ∼=
k∏
r=1
aut(Pr)Ssr .
In other words, the automorphism group of G is generated by the automorphism groups
of the factors and the transpositions of isomorphic factors.
Remark 3.2.2. (A quick verification of Theorem 3.2.1, and more.) Note that the
P srr , for r = 1, 2, . . . , k, are pairwise relatively prime. With Theorem 3.1.6 handy, we
see that Theorem 3.2.1 is equivalent to the following special case:
If P is any prime graph and k is a nonnegative integer, then the auto-
morphism group of P k is the exponentiation group aut(P )Sk , i.e.,
aut(P k) = aut(P )Sk.
In fact, we quickly observe that aut(P )Sk is a subgroup of aut(P k), since every
element
(α, τ) = (α; τ(1), τ(2), . . . , τ(k)) ∈ aut(P )Sk
54
CHAPTER 3. PRIME GRAPHS
with α ∈ Sk and τ(i) ∈ aut(P ), for all i, gives rise to an automorphism of the product
graph P k by letting each τ(i) act on a copy of P in P k and letting α act on the k
copies of P in P k. More precisely, since each of the vertices of P k is a k-vector where
the i-th coordinate is contributed by a vertex in the i-th copy of P , the permutation
α acts on P k by permuting the coordinates of the vertices of P k. Therefore, the
exponentiation group aut(P )Sk is a subgroup of the automorphism group of P k.
Hence what Sabidussi’s theorem really says is that the automorphism group of P k
is no bigger than the exponentiation group aut(P )Sk . Therefore, Theorem 3.2.1 can
be verified by showing that these two groups contain the same number of elements.
That is, it remains to show that
aut(P k) = | aut(P )Sk |.
Recall that for any graph G, kG is the sum of k copies of G. By Theorem 3.1.5,
we have
| aut(P k)| = | aut(kP )|.
Therefore, it suffices for us to show that
| aut(kP )| = k! | aut(P )|k,
where the right-hand side is the same as the order of the exponentiation group
aut(P )Sk .
Note that kP has k connected components, each of which is a copy of P . Let
α be an automorphism of kP , and let v1, v2 be vertices of kP that are in the same
connected component. Since automorphisms preserve adjacency, α(v1) and α(v2) are
in the same connected component of α(kP ) = kP . Therefore, α sends one connected
component of kP to another connected component of kP , possibly the same one.
55
CHAPTER 3. PRIME GRAPHS
S3
Figure 3.3. The automorphisms of kP are generated by elements inaut(P ) and Sk.
Let us assume, say, that α(P1) = P2, where P1 and P2 are two arbitrary connected
components of kP , possibly the same one. This means that P1 and P2 are both copies
of P . Let α1 be the action of α restricted to P1. Then there is some σ : P2 → P1
such that σα1 is an automorphism of P . Therefore, such an α can be obtained by
taking an automorphism of each connected component of kP , separately, followed by
putting a permutation on [k] over the k copies of P . In the meanwhile, the above
argument also implies that this is the only way to construct an automorphism of kP .
Thus we see from Definition 1.1.4 that the automorphism group of kP is the wreath
product group aut(P ) ≀ Sk, i.e.,
aut(kP ) = aut(P ) ≀ Sk,
which gives that
| aut(kP )| = | aut(P ) ≀ Sk| = k! | aut(P )|k.
In what follows in this section all graphs considered are connected.
Definition 3.2.3. The Dirichlet exponential generating series for a sequence of num-
bers ann∈N is defined by∑n≥1
an/n! ns.
56
CHAPTER 3. PRIME GRAPHS
Multiplication of Dirichlet exponential generating series is given by
(3.2.1)
(∑
n≥1
an
n! ns
)(∑
n≥1
bn
n! ns
)=
∑
n≥1
cn
n! ns,
where
cn =∑
k|n
n
k
akbn/k =
∑
k|n
n!
k! (n/k)!akbn/k.
The Dirichlet exponential generating function for a species F with the restriction
F [∅] = ∅ is defined by
D(F ) =∑
n≥1
|F [n]|
n! ns.
Example 3.2.4. Recall that for any graph G, OG is the species associated to G.
We denote by D(G) the Dirichlet exponential generating function for the species OG.
More explicitly,
D(G) = D(OG) =L(G)
l(G)! l(G)s.
It is illustrated by the following proposition that the Dirichlet exponential gener-
ating functions are useful for enumeration involving the arithmetic product of species.
Proposition 3.2.5. (Maia and Mendez) Let F1 and F2 be species with Fi[∅] = ∅
for i = 1, 2. Then
(3.2.2) D(F1 ⊡ F2) = D(F1) D(F2).
Proof. This proof was given by Maia and Mendez [18, p. 6]. We calculate the
number of F1 ⊡ F2-structures on the set [n]:
|F1 ⊡ F2[n]| =∑
(π1,π2)∈N [n]
|F1[π1]| |F2[π2]|
57
CHAPTER 3. PRIME GRAPHS
=∑
a|n
∑
(π1,π2)∈N [n]height(π1,π2)=a
|F1[a]| |F2[n/a]|
=∑
a|n
n
a
|F1[a]| |F2[n/a]|.
Equation (3.2.2) follows from the multiplication rule of Dirichlet exponential gen-
erating functions given by Equation (3.2.1).
Lemma 3.2.6. Let G1 and G2 be relatively prime graphs. Then
D(G1 ⊙ G2) = D(G1)D(G2)
Proof. Applying Proposition 3.1.8 and Proposition 3.2.5, we get
D(G1 ⊙ G2) = D(OG1⊙G2) = D(OG1 ⊡ OG2) = D(OG1)D(OG2) = D(G1)D(G2).
Lemma 3.2.7. Let P be any prime graph. Let T be the sum of all nonnegative
integer powers of P , i.e., T =∑
k≥0 P k. Then the Dirichlet exponential generating
functions for T and P are related by
D(T ) = exp(D(P )).
Proof. For any graph G, we have
L(G) =l(G)!
|aut(G)|,
and
D(G) =L(G)
l(G)! · l(G)s=
1
|aut(G)| · l(G)s.
58
CHAPTER 3. PRIME GRAPHS
Now it follows from Remark 3.2.2 that
∣∣aut(P k)∣∣ =
∣∣aut(P )Sk∣∣ = k! · |aut(P )|k ,
which gives that
L(P k) =l(P k)!
| aut(P k)|=
l(P k)!
k! · |aut(P )|k.
Hence we get
D(P k) =L(P k)
l(P k)! · l(P k)s=
1
k! · |aut(P )|k · l(P )ks=
D(P )k
k!.
Summing up on k, we get
D(T ) =∑
k≥0
D(P )k
k!= exp(D(P )).
Example 3.2.8. Let P be the species of prime graphs, and G c the species of con-
nected graphs. Then D(G c) and D(P) are the Dirichlet exponential generating
functions for these two species, respectively:
D(G c) =∑
n≥1
|G c[n]|
n! ns=
∑
G∈C
D(G), D(P) =∑
n≥1
|P[n]|
n! ns=
∑
P∈P
D(P ),
where C is the set of unlabeled connected graphs and P is the set of unlabeled prime
graphs.
Theorem 3.2.9. Let G c be the species of connected graphs, and let P be the
species of prime graphs. We have
D(G c) = exp (D(P)).
59
CHAPTER 3. PRIME GRAPHS
Proof. Lemma 3.2.6 gives that the Dirichlet exponential generating function of
a product of two relatively prime graphs is the product of the Dirichlet exponential
generating functions of the two graphs. Note that according to Proposition 3.1.2, the
operation of Cartesian product on graphs is associative up to isomorphism. Then it
follows that if we have a set of pairwise relatively prime graphs Gii=1,2,...,r, and let
G =r⊙i=1
Gi, then
D(G) =
r∏
i=1
D(Gi).
Now according to the definition of the Dirichlet exponential generating function
for graphs, we get
D(G c) =∑
G∈C
D(G) =∏
P∈P
D
(∑
k≥0
P k
)=
∏
P∈P
exp(D(P ))
= exp(∑
P∈P
D(P ))
= exp(D(P)).
Recall the exponential generating series of G c given by (2.4.5):
Gc (x) =
x
1!+
x2
2!+ 4
x3
3!+ 38
x4
4!+ 728
x5
5!+ 26704
x6
6!+ 1866256
x7
7!
+ 251548592x8
8!+ 66296291072
x9
9!+ . . . ,
We obtain D(G c) by replacing xn with n−s for each n in the above expression:
D(G c) =∑
n≥1
|G c[n]|1
n! ns
=1
1! 1s+
1
2! 2s+ 4
1
3! 3s+ 38
1
4! 4s+ 728
1
5! 5s+ 26704
1
6! 6s+ 1866256
1
7! 7s
+ 2515485921
8! 8s+ 66296291072
1
9! 9s+ . . . .
60
CHAPTER 3. PRIME GRAPHS
Theorem 3.2.9 gives a way of counting labeled prime graphs by writing
D(P) = log D(G c).
For example, we write down the first terms of D(P) as follows:
D(P) =1
1! 1s+
1
2! 2s+ 4
1
3! 3s+ 35
1
4! 4s+ 728
1
5! 5s+ 26464
1
6! 6s+ 1866256
1
7! 7s
+ 2515183521
8! 8s+ 66296210432
1
9! 9s+ . . . .
Let prn and cn be the number of labeled prime graphs and the number of labeled
connected graphs, respectively. We obtain the following table.
Table 1. Values of prn and cn, for n ≤ 12.
n prn cn
1 1 12 1 13 4 44 35 385 728 7286 26464 267047 1866256 18662568 251518352 2515485929 66296210432 66296291072
10 34496477587456 3449648859481611 35641657548953344 3564165754895334412 73354596197458024448 73354596206766622208
3.3. Unlabeled Prime Graphs
In this section all graphs considered are unlabeled and connected.
Definition 3.3.1. The (formal) Dirichlet series of a sequence ann=1,2,...,∞ is de-
fined to be∑∞
n=1 an/ns.
61
CHAPTER 3. PRIME GRAPHS
The multiplication of Dirichlet series is given by
∑
n≥1
an
ns·∑
m≥1
bn
ns=
∑
n≥1
(∑
k|n
akbn/k
)1
ns.
Definition 3.3.2. A monoid is a semigroup with a unit. A free commutative monoid
is a commutative monoid M with a set of primes P ⊆ M such that each element m ∈
M can be uniquely decomposed into a product of elements in P up to rearrangement.
Let M be a free commutative monoid. We get a monoid algebra CM , in which
the elements are all formal sums∑
m∈M cmm, where cm ∈ C, with addition and
multiplication defined naturally.
For each m ∈ M , we associate a length l(m) that is compatible with the multipli-
cation in M . That is, for any m1, m2 ∈ M , we have l(m1)l(m2) = l(m1m2).
It is well-known that the monoid algebra yields the following identity:
Proposition 3.3.3. Let M be a free commutative monoid with prime set P . The
following identity holds in the monoid algebra CM :
∑
m∈M
m =∏
p∈P
1
1 − p.
Furthermore, we can define a homomorphism from M to the ring of Dirichlet series
under which each m ∈ M is sent to 1/l(m)s, where l is a length function of M .
Therefore,∑
m∈M
1
l(m)s=
∏
p∈P
1
1 − l(p)−s.
Example 3.3.4. Let N denote the set of all natural numbers, and let P denote the
set of all prime numbers. Then N is a free commutative monoid with prime set P,
and the length function is given by l(n) = n for all n ∈ N. As an application of
Proposition 3.3.3, we have the following well-known identity for expressing the zeta
62
CHAPTER 3. PRIME GRAPHS
function:
ζ(s) =∑
n∈N
1
ns=
∏
p∈P
1
1 − p−s.
Recall that C is the set of unlabeled connected graphs under the operation of
Cartesian product. The unique factorization theorem of Sabidussi gives C the struc-
ture of a commutative free monoid with a set of primes P, where P is the set of
unlabeled prime graphs. This is saying that every element of C has a unique fac-
torization of the form be11 be2
2 · · · bek
k , where the bi are distinct primes in P. Let l(G),
the number of vertices in G, be a length function for C. We have the following
proposition.
Lemma 3.3.5. Let C and P be the set of unlabeled connected graphs and the set of
unlabeled prime graphs, respectively. We have
∑
G∈C
1
l(G)s=
∏
P∈P
1
1 − l(P )−s.
The enumeration of prime graphs was studied by Raphael Bellec [1]. We use
Dirichlet series to count unlabeled connected prime graphs.
Theorem 3.3.6. Let cn be the number of unlabeled connected graphs on n vertices,
and let bm be the number of unlabeled prime graphs on m vertices. Then we have
(3.3.1)∑
n≥1
cn
ns=
∏
m≥2
1
(1 − m−s)bm.
Furthermore, if we define numbers dn for positive integers n by
(3.3.2)∑
n≥1
dn
ns= log
∑
n≥1
cn
ns,
63
CHAPTER 3. PRIME GRAPHS
then
(3.3.3) dn =∑
ml=n
bm
l,
where the sum is over all pairs (m, l) of positive integers with ml = n.
Remark 3.3.7. A quick observation from Equation (3.3.3) is that bn = dn whenever
n is not of the form rk for some k > 1.
In what follows, we introduce an interesting recursive formula for computing dn.
To start with, we differentiate both sides of Equation (3.3.2) with respect to s and
simplify. We get that
∑
n≥2
log ncn
ns=
(∑
n≥1
cn
ns
)(∑
n≥2
log ndn
ns
),
which gives
(3.3.4) cn log n =∑
ml=n
cmdl log l.
Since c1 is the number of connected graphs on 1 vertex, c1 = 1. It follows from
Equation (3.3.4) easily that dp = cp when p is a prime number. Therefore, if p is a
prime number, bp = dp = cp. This fact can be seen directly, since a connected graph
with a prime number of vertices is a prime graph.
Raphael Bellec used Equation (3.3.4) to find formulae for dn where n is a product
of two different primes or a product of three different primes:
If n = pq where p 6= q,
(3.3.5) dn = cn − cpcq;
64
CHAPTER 3. PRIME GRAPHS
If n = pqr where p, q and r are distinct primes,
(3.3.6) dn = cn + 2cpcqcr − cpcqr − cqcpr − crcpq.
In fact, Equations (3.3.5) and (3.3.6) are special cases of the following proposition.
Proposition 3.3.8. Let dn, cn be defined as above. Then we have
dn = cn −1
2
∑
n1n2=n
cn1cn2 +1
3
∑
n1n2n3=n
cn1cn2cn3 − . . . .
Proof. We can use the identity
log(1 + x) = x −1
2x2 +
1
3x3 −
1
4x4 + . . .
to compute from Equation (3.3.2) that
∑
n≥1
dn
ns= log
(1 +
∑
n≥2
cn
ns
)
=∑
n≥2
cn
ns−
1
2
(∑
n≥2
cn
ns
)2
+1
3
(∑
n≥2
cn
ns
)3
− . . . .
Equating coefficients of n−s on both sides, we get the desired result.
Proof of Theorem 3.3.6. We start with
∑
m
1
l(m)s=
∏
p
1
1 − l(p)−s,
where the left-hand side is summed over all connected graphs, and the right-hand side
is summed over all prime graphs. Regrouping the summands on the left-hand side with
respect to the number of vertices in m, we get the left-hand side of Equation (3.3.1).
Regrouping the factors on the right-hand side with respect to the number of vertices
in p, we get the right-hand side of Equation (3.3.1).
65
CHAPTER 3. PRIME GRAPHS
Taking the logarithm of both sides of Equation (3.3.1), we get
log∑
n≥1
cn
ns= log
∏
m≥2
1
(1 − m−s)bm=
∑
m≥2
bm log1
1 − m−s
=∑
m≥2
(bm
∑
l≥1
m−sl
l
)=
∑
m≥2, l≥1
bm
l msl,
and Equation (3.3.3) follows immediately.
Next, we will compute the numbers bn in terms of the numbers dn using the
following lemma.
Lemma 3.3.9. Let Dii=1,... and Jii=1,... be sequences of numbers satisfying
(3.3.7) Dk =∑
l|k
Jk/l
l,
and let µ be the Mobius function. Then we have
Jk =1
k
∑
l|k
µ
(k
l
)lDl.
Proof. Multiplying by k on both sides of Equation (3.3.7) , we get
kDk =∑
l|k
k
lJk/l =
∑
l|k
lJl.
Applying the Mobius inversion formula, we get
kJk =∑
l|k
µ
(k
l
)lDl.
Therefore,
Jk =1
k
∑
l|k
µ
(k
l
)lDl.
66
CHAPTER 3. PRIME GRAPHS
Given any natural number n, let e be the largest number such that n = re for
some r. Note that r is not a power of a smaller integer. We let Dk = drk, Jk = brk . It
follows that Equation (3.3.3) is equivalent to Equation (3.3.7).
Theorem 3.3.10. For any natural number n, let e, r be as described in above.
Then we have
bn =1
e
∑
l|e
µ
(e
l
)ldre.
Proof. The result follows straightforwardly from Lemma 3.3.9.
Table 2 gives the numbers of unlabeled prime graphs bn compared with the num-
bers of unlabeled connected graphs cn on no more than 12 vertices.
Table 2. Values of cn and bn, for n ≤ 12.
n cn bn
1 1 02 1 13 2 24 6 55 21 216 112 1107 853 8538 11117 111119 261080 261077
10 11716571 1171655011 1006700565 100670056512 164059830476 164059830354
3.4. Exponential Composition with a Molecular Species
In this section we introduce an operation on species that is equivalent to the
exponentiation groups in the case of molecular species. This operation turns out to
be useful for finding a functional equation relating the species of connected graphs
and the species of prime graphs.
67
CHAPTER 3. PRIME GRAPHS
Let F be a species of structures with F [∅] = ∅, let k be a positive integer, and let
A be a subgroup of Sk. Recall that an F ⊡k-structure on a finite set U is a tuple of
the form
((π1, f1), (π2, f2), . . . , (πk, fk)),
where (π1, π2, . . . , πk) is a k-rectangle on U , and each fi is an F -structure on the blocks
of πi. The group A acts on the set of F ⊡k-structures by permuting the subscripts of
πi and fi, i.e.,
α((π1, f1), . . . , (πk, fk)) = ((πα(1), fα(1)), . . . , (πα(k), fα(k))),
where α is an element of A, (πα1 , πα2 , . . . , παk) is a k-rectangle on U , and each fαi
is an F -structure on the blocks of παi. It is easy to check that this action of A on
F ⊡k-structures is natural, that is, it commutes with any bijection σ : U → V . Hence
we get a quotient species under this group action.
Definition 3.4.1. We define the exponential composition of F with the molecular
species Xk/A to be the quotient species, denoted (Xk/A)〈F 〉, under the group action
described in above. That is,
(Xk/A)〈F 〉 := F ⊡k/A.
Example 3.4.2. Figure 3.4 illustrates the group action of S2 on the set of E ⊡22 -
structures on [4], resulting in three orbits.
Let A be a subgroup of Sm, and let B be a subgroup of Sn. We have the following
formulas from Theorem 2.3.10 and Proposition 2.6.5:
Xm
A·Xn
B=
Xm+n
A > B,
68
CHAPTER 3. PRIME GRAPHS
1 2
43
1 3
2 4 4
1 3
23
1 4 1 4
32
21
4 32
Figure 3.4. There are three elements in the set E2〈E2〉[4].
Xm
A
Xn
B=
Xmn
B ≀ A,
Xm
A⊡
Xn
B=
Xmn
A × B.
Now we can add to this list one more formula representing the group action of BA
on the set N = nm stated in the following theorem.
Theorem 3.4.3. Let A and B be groups described in above, let N = nm, and let
BA be the exponentiation group of A with B, as defined in Definition 1.3.1. Then we
have
Xm
A
⟨Xn
B
⟩=
XN
BA.
Moreover, we get the cycle index of (Xm/A)〈Xn/B〉:
Z(Xm/A)〈Xn/B〉 = Z(A) ∗ Z(B),
where the expression Z(A) ∗ Z(B) denotes the image of Z(B) under the operator
obtained by substituting the operator Ir for the variables pr in Z(A).
69
CHAPTER 3. PRIME GRAPHS
Proof. Proposition 2.6.5 and the associativity of the arithmetic product give
that (Xn
B
)⊡m
=XN
Bm,
where Bm is the the product of m copies of B, acting on the set
[n] × [n] × · · · × [n]︸ ︷︷ ︸m copies
piecewise, and hence viewed as a subgroup of SN . According to Remark 2.3.4, the
set of (Xn/B)⊡m-structures on [N ] is then just the set of Bm-orbits of linear orders
on [N ].
The group A acts on these Bm-orbits of linear orders by permuting the subscripts.
This action results in the quotient species
Xm
A
⟨Xn
B
⟩=
(Xn
B
)⊡m/A =
(XN
Bm
)/A.
We observe that an A-orbit of Bm-orbits of linear orders on [N ] admits an au-
tomorphism group isomorphic to the exponentiation group BA, hence the quotient
species (XN/Bm)/A is the same as the molecular species XN/BA.
As for the cycle index of (Xm/A)〈Xn/B〉, we apply Proposition 2.3.5 and Theo-
rem 1.3.6 to get that
Z(Xm/A)〈Xn/B〉 = ZXN /BA = Z(BA) = Z(A) ∗ Z(B).
Figure 3.5 illustrates an group action of A on a set of (Xn/B)⊡m-structures.
Next, we introduce a theorem that is a generalization of Palmer and Robinson’s
Theorem 1.3.6.
70
CHAPTER 3. PRIME GRAPHS
A-orbits
B-orb
its
B-o
rbit
s
B-orbits B-orbits
B-orbits
A-orbits
A-o
rbit
sA-orb
its
A-orbits
Figure 3.5. ((Xn/B)⊡m)/A = Xnm
/BA.
Theorem 3.4.4. Let A be a subgroup of Sk, and let F be a species of structures
concentrated on the cardinality n. Then the cycle index of the species (Xk/A)〈F 〉 is
given by
(3.4.1) Z(Xk/A)〈F 〉 = Z(A) ∗ ZF .
Remark 3.4.5 (Notation and Set-up). We denote by Parn the set of partitions of
n, and by Parkn the set of k-sequences of partitions of n.
For fixed integers n, k, and N = nk, we denote by NN the species of k-dimensional
cubes, or k-cubes, on [N ], defined by
NN = E⊡kn [N ].
We also call the elements of the set (Xn)⊡k[N ] k-dimensional ordered cubes on [N ].
71
CHAPTER 3. PRIME GRAPHS
Let σ be a permutation on [k] with cycle type
c. t.(σ) = (r1, r2 . . . , rd).
Then σ acts on the F ⊡k-structures by permuting the subscripts. Let ν be a partition
of N . Let δ be a permutation of [N ] with cycle type ν. Then δ acts on the F ⊡k-
structures by transport of structures. Recall the notation introduced in Remark 1.3.8:
I(c. t.(σ); λ(1), λ(2), . . . , λ(d)) = Ir1(pλ(1)) ⊠ Ir2(pλ(2)) ⊠ · · ·⊠ Ird(pλ(d)).
We denote by RecF (σ, ν) a function on the pair (σ, ν) defined by
(3.4.2) RecF (σ, ν) :=∑ ∏d
i=1 fix F [λ(i)]
zλ(1) · · · zλ(d)
,
where the summation is over all sequences (λ(1), λ(2), . . . , λ(d)) in Pardn with
I(c. t.(σ); λ(1), λ(2), . . . , λ(d))) = pν .
We denote by fixF (σ, δ) the number of F ⊡k-structures on the set [N ] fixed by the
joint action of the pair (σ, δ).
Proof of Theorem 3.4.4. Let ν be a partition of N . It suffices to prove that
the coefficients of pν on both sides of Equation (3.4.1) are equal.
The right-hand side of Equation (3.4.1) is
Z(A) ∗ ZF =
(1
|A|
∑
σ∈A
pc.t.(σ)
)∗
(∑
λ⊢n
fix F [λ]pλ
zλ
)
=1
|A|
∑
σ∈A
Ic.t.(σ)
(∑
λ⊢n
fix F [λ]pλ
zλ
).
72
CHAPTER 3. PRIME GRAPHS
For σ ∈ A with c. t.(σ) = (r1, r2, . . . , rd), we have
Ic.t.(σ) = Ir1 · · · Ird,
and
Ic.t.(σ)
(∑
λ⊢n
fix F [λ]pλ
zλ
)
= Ir1
(∑
λ⊢n
fix F [λ]pλ
zλ
)⊠ Ir2
(∑
λ⊢n
fix F [λ]pλ
zλ
)⊠ · · · ⊠ Ird
(∑
λ⊢n
fix F [λ]pλ
zλ
).
Therefore, the coefficient of pν in the expression Z(A) ∗ ZF is
1
|A|
∑ ∏di=1 fix F [λ(i)]
zλ(1) · · · zλ(d)
=1
|A|
∑
σ∈A
RecF (σ, ν),(3.4.3)
where the summation on the left-hand side is taken over all sequences (λ(1), λ(2), . . . , λ(d))
in Pardn for some d ≥ 1 and all σ ∈ A with c. t.(σ) = (r1, r2, . . . , rd) such that
I(c. t.(σ); λ(1), λ(2), . . . , λ(d)) = pν ,
and RecF (σ, ν) on the right-hand side is as defined by (3.4.2) in Remark 3.4.5.
The left-hand side of Equation (3.4.1) is
ZF ⊡k/A =∑
ν⊢N
fixF ⊡k
A[ν]
pν
zν
.
Therefore, the coefficient of pν in the expression ZF ⊡k/A is
1
zνfix
F ⊡k
A[ν].
73
CHAPTER 3. PRIME GRAPHS
We then apply Theorem 1.1.6 to get that the number of A-orbits of F ⊡k-structures
on [N ] fixed by a permutation δ ∈ SN of cycle type ν is
fixF ⊡k
A[ν] = fix
F ⊡k
A[δ] =
1
|A|
∑
σ∈A
fixF (σ, δ),(3.4.4)
where fixF (σ, δ) is as defined in Remark 3.4.5.
Therefore, combining (3.4.4) and (3.4.3), the proof of Equation (3.4.1) is reduced
to showing that
(3.4.5) fixF (σ, δ) = zν RecF (σ, ν),
for any δ, ν and σ.
To prove (3.4.5), we start with observing that in order for an F ⊡k-structure on
[N ] of the form
((π1, f1), (π2, f2), . . . , (πk, fk))
to be fixed by the pair (σ, δ), it is necessary that (σ, δ) fixes the k-cube of the form
(π1, π2, . . . , πk) ∈ NN . This is equivalent to saying that
(3.4.6) NN [δ](π1, π2, . . . , πk) = (πσ(1), πσ(2), . . . , πσ(k)).
Suppose (3.4.6) holds for some k-cube (π1, π2, . . . , πk) ∈ NN . We let βi ∈ Sn be
the induced action of δ on the blocks of πi, for i = 1, 2, . . . , k. That is,
NN [δ](πi) = βi(πσ(i))
for all i ∈ [k].
Now we consider the simpler case when σ is a k-cycle, say, σ = (1, 2, . . . , k). Then
the action of δ sends (π1, π2, . . . , πk) to (π2, π3, . . . , π1). Let β = β1β2 · · ·βk. The
74
CHAPTER 3. PRIME GRAPHS
πσ(2)
β2 βkβ1
π1 π2 πk
πσ(k)πσ(1)
above discussion is saying that, according to Remark 1.3.3,
Ik(pc.t.(β)) = pν .
On the other hand, given a partition λ of n satisfying Ik(pλ) = pν , there are n!/zλ
permutations in Sn with cycle type λ. Let β be one of such. Then the number of
sequences (β1, β2, . . . , βk) whose product equals β is (n!)k−1, since we can choose β1 up
to βk−1 freely, and βk is thereforee determined. All such sequences (β1, β2, . . . , βk) will
satisfy Ik(pc.t.(β1···βk)) = pν , thus their action on an arbitrary k-dimensional ordered
cube, combined with the action of σ on the subscripts, would result in a permutation
on [N ] with cycle type ν. But there are N !/zν permutations with cycle type ν, and
only one of them is the δ that we started with. Considering that the k-cubes are just
Skn-orbits of the k-dimensional ordered cubes, we count the number of k-cubes that
are fixed by the pair (σ, δ) with the further condition that the product of the induced
permutations on the πi by δ has cycle type λ:
#
(β1,β2,...,βk)∈Skn
with c.t.(β1···βk)=λ
· #
k-dimensional ordered cubes
#
permutations on [N ]with cycle type ν
· #
k-dimensional ordered cubes
in each equivalence class
=[(n!)k−1 · n!/zλ] · N !
N !/zν · (n!)k=
zν
zλ
.
75
CHAPTER 3. PRIME GRAPHS
Now we try to compute how many F ⊡k-structures of the form
((π1, f1), (π2, f2), . . . , (πk, fk)),
based on a given rectangle (π1, π2, . . . , πk) that is fixed by the βi with
c. t.
(∏
i
βi
)= λ,
are fixed by the pair (σ, δ). We observe that the action of (σ, δ) determines that
fk = F [β1]f1 and fi = F [βi−1]fi−1 for i = 2, 3, . . . , k, and hence
fk = F [β1]F [β2] · · ·F [βk]fk = F [β]fk = F [λ]fk.
In other words,
fk ∈ Fix F [λ].
Hence as long as we choose an fk from Fix F [λ], then all the other fi for i < k are
determined by our choice of fk. There are fix F [λ] such choices for fk.
Therefore, in the case when σ is a k-cycle, we get that the number of F ⊡k-
structures on the set [N ] fixed by the pair (σ, δ) is
fixF (σ, δ) =∑
λ⊢nIk(pλ)=pν
fix F [λ]zν
zλ= zν RecF (σ, ν).
Now let us consider the general case when σ contains d cycles of lengths r1, r2, . . . , rd.
Let (π1, π2, . . . , πk) be a k-cube fixed by the pair (σ, δ). Again we have (3.4.6), and
we get an induced βi on the blocks of πσ−1i for each i.
We observe that the action of σ on the subscripts of the k-cube partitions the
list π1, π2, . . . , πk into d parts, of lengths r1, r2, . . . , rd, within each of which we get a
ri-cycle. We group the βi on each of the d parts and get d permutations in the group
76
CHAPTER 3. PRIME GRAPHS
Sn, whose cycle types are denoted by λ(1), λ(2), . . . , λ(d). This construction gives that
such a sequence of partitions (λ(1), λ(2), . . . , λ(d)) will be those that satisfy
I(c. t.(σ); λ(1), λ(2), . . . , λ(d)) = pν .
Therefore, the number of k-cubes fixed by (σ, δ) corresponding to such a sequence
of partitions (λ(1), λ(2), . . . , λ(d)) is
(n!)r1−1 · n!/zλ(1) · · · · · (n!)rd−1 · n!/zλ(d)
N !/zν·
N !
(n!)k
=(n!)r1+···+rd
N !
zν
zλ(1) · · · zλ(d)
=zν
zλ(1) · · · zλ(d)
.
The number of F -structures that are assigned to this k-cube (π1, π2, . . . , πk) that
will be fixed under the action of the pair (σ, δ) corresponding to the sequence of
partitions (λ(1), λ(2), . . . , λ(d)) is hence
fix F [λ(1)] · · ·fix F [λ(d)],
since, similarly to our previous discussion, within each of the d parts, we only need
to pick an F -structure that is fixed by a permutation of cycle type λ(i), and all other
F -structures are left determined.
Therefore, we get that for any pair (σ, δ),
fixF (σ, δ) = zν RecF (σ, ν),
which concludes our proof.
Remark 3.4.6. We can use the molecular decomposition to define the exponential
composition of a species F with a species H . That is, if the molecular decomposition
77
CHAPTER 3. PRIME GRAPHS
of H is given by
H =∑
M⊆HM molecular
M,
then we define H〈F 〉 by
H〈F 〉 =∑
M⊆HM molecular
M〈F 〉.
The left-linearity of the operation ∗ gives that the cycle index of H〈F 〉 is
ZH〈F 〉 = ZH ∗ ZF
=
( ∑
M⊆HM molecular
ZM
)∗ ZF
=∑
M⊆HM molecular
ZM ∗ ZF .
3.5. Exponential Composition
Let F be a species with F [∅] = ∅, and let k be a positive integer. We call the
quotient species Ek〈F 〉 = F ⊡k/Sk the exponential composition of F of order k. We
observe that an Ek〈F 〉-structure on a finite set U is a set of the form
(π1, f1), (π2, f2), . . . , (πk, fk),
where (π1, π2, . . . , πk) is a k-rectangle on U , and each fi is an F -structure on the
blocks of πi. We set E0〈F 〉 = X.
Definition 3.5.1. Let F be a species with F [∅] = F [1] = ∅. We define the expo-
nential composition of F , denoted E 〈F 〉, to be the sum of Ek〈F 〉 on all nonnegative
integers k, i.e.,
E 〈F 〉 =∑
k≥0
Ek〈F 〉.
78
CHAPTER 3. PRIME GRAPHS
Proposition 3.5.2. Let F be a species with F [∅] = ∅. Then the cycle index of
the exponential composition of F is given by
ZE 〈F 〉 = p1 +∑
k≥1
Z(Sk) ∗ ZF .
Proof. We get from Theorem 3.4.4 that for k ≥ 1,
ZEk〈F 〉 = Z(Sk) ∗ ZF .
And the rest follows from the partial linearity of the operation ∗ on symmetric func-
tions.
The exponential composition has the following properties.
Theorem 3.5.3. Let F1 and F2 be species with F1[∅] = F2[∅] = ∅, and let k be
any nonnegative integer. Then
Ek〈F1 + F2〉 =
k∑
i=0
Ei〈F1〉 ⊡ Ek−i〈F2〉,
E 〈F1 + F2〉 = E 〈F1〉 ⊡ E 〈F2〉.(3.5.1)
We observe that an (F1 +F2)⊡k-structure on a finite set U is a rectangle on U with
each partition in the rectangle enriched with either an F1 or an F2-structure. Taking
the Sk-orbits of these (F1 + F2)⊡k-structures on U means basically making every
partition of the rectangle “indistinguishable”. Hence in each Sk-orbit, all partitions
enriched with an F1-structure are grouped together to give an Sk1-orbit of F ⊡k11 -
structures, and the remaining partitions are grouped together to give an Sk2-orbit of
F ⊡k22 -structures, where k1 and k2 are nonnegative integers whose sum is equal to k.
79
CHAPTER 3. PRIME GRAPHS
Proof of Theorem 3.5.3. First, we prove that for any nonnegative integer k,
Ek〈F1 + F2〉 =k∑
i=0
Ei〈F1〉 ⊡ Ek−i〈F2〉.
The case when k = 0 is trivial. Let us consider k to be a positive integer. Let s
and t be nonnegative integers whose sum equals k. Let U be a finite set. To get an
Es〈F1〉 ⊡ Et〈F2〉-structure on U , we first take a rectangle (ρ, τ) on U , and then take
an ordered pair (a, b), where a is an Es〈F1〉-structure on the blocks of ρ, and b is an
Et〈F2〉-structure on the blocks of τ . That is,
a = (ρ1, f1), . . . , (ρs, fs), b = (τ1, g1), . . . , (τt, gt),
where (ρ1, . . . , ρs) is a rectangle on the blocks of ρ, (τ1, . . . , τt) is a rectangle on the
blocks of τ , fi is an F1-structure on the blocks of ρi, and gj is an F2-structure on the
blocks of τj .
Recall that Proposition 2.6.8 says for any nonnegative integers i, j,
N(i+j) = N
(i)⊡ N
(j).
It follows that (ρ1, . . . , ρs, τ1, . . . , τt) is a rectangle on U .
On the other hand, let x be an Ek〈F1 + F2〉-structure on U . We can write x as a
set of the form
x = (π1, f1), . . . , (πr, fr), (πr+1, gr+1), . . . , (πk, gk),
where (π1, π2, . . . , πk) is a k-rectangle on U , r is a nonnegative integer between 0 and
k, each fi is an F1-structure on πi for i = 1, . . . , r, and each gj is an F2-structure on
πj for j = r + 1, . . . , k.
80
CHAPTER 3. PRIME GRAPHS
We then write x = (x1, x2), where
x1 = (π1, f1), . . . , (πr, fr), x2 = (πr+1, gr+1), . . . , (πk, gk).
Hence running through values of s and t, we get that the set of Es〈F1〉 ⊡ Et〈F2〉-
structures on U , written in the form of the pairs (a, b) whose construction we described
in above, corresponds naturally to the set of Ek〈F1 + F2〉-structures on U .
The proof of
E 〈F1 + F2〉 = E 〈F1〉 ⊡ E 〈F2〉.
is straightforward using the properties of the arithmetic product, namely, the com-
mutativity, associativity and distributivity:
E 〈F1 + F2〉 =∑
k≥0
Ek〈F1 + F2〉
=∑
k≥0
∑
i+j=ki,j≥0
Ei〈F1〉 ⊡ Ej〈F2〉
=
(∑
i≥0
Ei〈F1〉
)⊡
(∑
j≥0
Ej〈F2〉
)
= E 〈F1〉 ⊡ E 〈F2〉.
Note that identity (3.5.1) is analogous to the identity about the composition of a
sum of species with the species of sets E :
E (F1 + F2) = E (F1) E (F2).
What is more, (3.5.1) illustrates a kind of distributivity of the exponential composi-
tion. In fact, if a species of structures F has its molecular decomposition written in
81
CHAPTER 3. PRIME GRAPHS
the form
F =∑
M⊆FM molecular
M,
then the exponential composition of F can be written as
E 〈F 〉 = ⊡M⊆F
M molecular
E 〈M〉.
3.6. Cycle Index of Prime Graphs
Now we are ready to come back to the species of prime graphs.
Lemma 3.6.1. Let P be any prime graph, and k any nonnegative integer. Then
the species associated to the k-th power of P is the exponential composition of OP of
order k. That is,
OP k = Ek〈OP 〉.
Proof. We apply Theorem 3.4.3 and get
Ek〈OP 〉 = O⊡kP /Sk =
(Xn
aut(P )
)⊡k/Sk =
Xnk
aut(P )Sk.
It follows from Remark 3.2.2 that
Ek〈OP 〉 =Xnk
aut(P k)= OP k.
We can verify Lemma 3.6.1 in an intuitive way. Note that the set of Ek〈OP 〉-
structures on a finite set U is the set of Sk-orbits of O⊡kP -structures on U , and an
element of Ek〈OP 〉[U ] of the form (π1, f1), . . . , (πk, fk) is such that (π1, π2, . . . , πk)
is a k-rectangle on U , and each fi is a graph isomorphic to P whose vertex set
equal to the blocks of πi. Such a set (π1, f1), . . . , (πk, fk) corresponds to a graph G
82
CHAPTER 3. PRIME GRAPHS
isomorphic to P k with vertex set U . More precisely, G is the Cartesian product of
the fi in which each vertex u ∈ U is of the form u = B1 ∩ B2 ∩ · · · ∩ Bk, where each
Bi is one of the blocks of πi. In this way, we get a one-to-one correspondence between
the Ek〈OP 〉-structures on U and the set of graphs isomorphic to P k with vertex set
U .
Theorem 3.6.2. The species G c of connected graphs and P of prime graphs
satisfy
Gc = E 〈P〉.
Proof. In this proof, all graphs considered are unlabeled.
The molecular decomposition of the species of prime graphs is
P =∑
P prime
OP ,
where each OP is a molecular species which is isomorphic to X l(P )/ aut(P ).
Let P1, P2, . . . be the set of unlabeled prime graphs. We have
E 〈P〉 = E 〈OP1 + OP2 + · · · 〉
= E 〈OP1〉 ⊡ E 〈OP2〉 ⊡ · · ·
= (X + OP1 + OP 21
+ · · · ) ⊡ (X + OP2 + OP 22
+ · · · ) ⊡ · · ·
=∑
i1,i2,···≥0
OP
i11
⊡ OP
i22
⊡ · · ·
=∑
i1,i2,···≥0
OP
i11 ⊡P
i22 ⊡···
=∑
C connected
OC
= Gc.
83
CHAPTER 3. PRIME GRAPHS
Now we can compute the cycle index of the species of prime graphs recursively
using Maple:
ZP =
(1
2p2
1 +1
2p2
)+
(2
3p3
1 + p1p2 +1
3p3
)
+
(35
24p4
1 +7
4p2
1p2 +2
3p1p3 +
7
8p2
2 +1
4p4
)
+
(91
15p5
1 +19
3p3
1p2 +4
3p3
1p3 + 5p1p22 + p1p4 +
2
3p2p3 +
3
5p5
)
+
(1654
45p6
1 +91
3p4
1p2 +38
9p3
1p3 + 21p21p
22 + 2p2
1p4 +8
3p1p2p3
+4
5p1p5 +
47
6p3
2 +5
2p2p4 +
11
9p2
3 +2
3p6
)+ · · · .
Figure 3.6. Unlabeled prime graphs on n vertices, n ≤ 4.
We can write down the beginning terms of the molecular decomposition of the
species P:
P = E2 + (XE2 + E3) + (E2 X2 + XE3 + X2E2 + E
22 + E4) + · · · .
Comparing Figure 3.6 with Figure 2.9, we see that there is only one unlabeled
connected graph with 4 vertices that is not prime. In fact, if we compare the first
84
CHAPTER 3. PRIME GRAPHS
several terms of ZG c , given in (2.4.6), and ZP of order no more than 6, we get that
ZG c − ZP = p1 +1
8
(p4
1 + 2p21p2 + 3 p2
2 + 2p4
)
+1
4
(p6
1 + p21p
22 + 2p3
2
)+
1
12
(p6
1 + 3p21p
22 + 4p3
2 + 2p23 + 2p6
)· · · ,
which is the cycle index of connected non-prime graphs on no more than 6 vertices,
as shown in Figure 3.7, which consist of a single vertex, a graph with 4 vertices, and
Figure 3.7. Unlabeled non-prime graphs on n vertices, n ≤ 6.
two graphs with 6 vertices.
85
CHAPTER 4
Point-Determining Graphs
4.0. Introduction
In this chapter, we examine the species of point-determining graphs (Definition 4.1.1),
which are graphs whose vertices all have distinct neighborhoods, the species of co-
point-determining graphs (Definition 4.1.2), which are graphs whose complements
are point-determining, the species of connected point-determining graphs, and the
species of bi-point-determining graphs (Definition 4.3.1), which are graphs that are
both point-determining and co-point-determining. Our goal is to find the cycle indices
of these species. First, we find a functional equation relating the species of point-
determining graphs and the well-known species of graphs (Theorem 4.1.3). At the
same time, we find a simple connection between the point-determining graphs and the
co-point-determining graphs. The connected cases are similar to the enumeration of
connected graphs as shown in Section 2.4. Furthermore, we find a functional equation
relating the species of bi-point-determining graphs and the species of graphs.
In addition, we examine the 2-sort species of 2-colored graphs (Definition 4.4.1),
which are graphs whose vertices are properly colored with white and black. We
develop ways to enumerate the 2-sort species of connected 2-colored graphs and the
2-sort species of point-determining 2-colored graphs.
86
CHAPTER 4. POINT-DETERMINING GRAPHS
4.1. Point-Determining Graphs and Co-Point-Determining Graphs
Definition 4.1.1. A point-determining graph, previously studied by Sumner [28],
also called a mating-type graph by Bull and Pease [4] and Read [23], is a graph G
in which any two distinct vertices have distinct neighborhoods, i.e., if v1 6= v2, then
N(v1) 6= N(v2), where N(v) = w : wv is an edge of G is the set of vertices adjacent
to v.
Note that the neighborhood of an isolated vertex is the empty set. Thus in a
point-determining graph, there is at most one isolated vertex.
Definition 4.1.2. A graph is called co-point-determining if its complement is point-
determining.
In a co-point-determining graph, two non-adjacent distinct vertices have distinct
augmented neighborhoods. That is, if v1 and v2 are distinct vertices in this graph,
then N(v1) ∪ v1 6= N(v2) ∪ v2.
We denote the species of point-determining graphs by P. Thus in the species
language, a point-determining graph is a P-structure. We denote by Q the species
of co-point-determining graphs. We set the number of point-determining graphs and
the number of co-point-determining graphs on an empty set of vertices both to be
one.
We observe that there is a natural transformation α that produces for every finite
set U a bijection between P[U ] and Q[U ], which sends each point-determining graph
with vertex set U to its complement, which is a co-point-determining graph with
vertex set U . Furthermore, the following diagram commutes for any finite sets U , V
and any bijection σ : U → V :
87
CHAPTER 4. POINT-DETERMINING GRAPHS
P[U ]P[σ]−−−→ P[V ]
α
yyα
Q[U ]Q[σ]
−−−→ Q[V ]Hence, according to Remark 2.1.7, the species P and Q are isomorphic, and we
do not make distinctions between them during calculations.
There are four point-determining graphs and and four co-point-determining graphs
labeled on [3], as shown in Figure 4.1, whose first row consists of the point-determining
graphs and whose second row consists of the co-point-determining graphs.
1
32
1
32
1
32
1
32
1
32
1
32
1
32
1
32
Figure 4.1. Labeled point-determining graphs and co-point-determining graphs with vertex set [3].
The following theorem gives a starting point for counting point-determining graphs.
Theorem 4.1.3. Let G be the species of graphs, P the species of point-determining
graphs, and E+ the species of nonempty sets. Then we have
(4.1.1) G = P(E+).
Proof. Given any graph G with vertex set V , we define an equivalence relation
on V by setting two elements v and w of V to be equivalent to each other if they
88
CHAPTER 4. POINT-DETERMINING GRAPHS
have the same neighborhoods. This gives a partition of V into m equivalence classes
π = V1, V2, . . . , Vm.
We then associated to the graph G the ordered pair (π, G′), where G′ is the graph
whose vertices are the blocks of π in which two vertices Vi and Vj in G′ are adjacent if
and only if there is an edge in G connecting a vertex in Vi with one in Vj . An example
of such a transformation from a graph G to a point-determining graph G′ is illustrated
in Figure 4.2. It is straightforward to see that the graph G′ is point-determining, since
no two vertices of G′ have the same neighborhoods.
4
3
1
9 2
6
5
74
8
3
1
9 2
6
5
7
8
Figure 4.2. A transformation from a graph G with vertex set [11]to a point-determining graph G′ with vertex set1, 9, 3, 8, 4, 7, 6, 2, 5.
On the other hand, we can construct a graph on n vertices uniquely (up to isomor-
phism) from an ordered pair consisting of a point-determining graph on m vertices
and a partition of n vertices into m blocks by reversing the procedure described in
above. This bijection leads to the species equivalence relation (4.1.1).
89
CHAPTER 4. POINT-DETERMINING GRAPHS
Using the same method we can prove that
(4.1.2) G (X) = Q(K+),
where K+ is the species of complete graphs with non-empty vertex sets.
Recall Γ, the compositional inverse of E+ as defined in Section 2.4. Equations (4.1.1)
and (4.1.2) can be rewritten as
P = Q = G (Γ).
This identity gives rise to several identities that can be used to compute the associated
series of P:
P(x) = Q(x) = G (log(1 + x)),(4.1.3)
P(x) = Q(x) = ZG (x − x2, x2 − x4, . . . ),(4.1.4)
ZP = ZQ = ZG
(∑
k≥1
µ(k)
klog(1 + pk),
∑
k≥1
µ(k)
klog(1 + p2k), . . .
).
Read [23] derived formulas (4.1.3) and (4.1.4), and pointed out that identity (4.1.3)
gives an explicit expression for the numbers of labeled point-determining graphs with
k vertices pk:
pk =∑
n≥0
2(n2 ) s(n, k),
where the s(n, k) denote Stirling numbers of the first kind [29].
Using Maple, we can write down the first several terms of the associated series of
P:
P(x) = 1 +x
1!+
x2
2!+ 4
x3
3!+ 32
x4
4!+ 588
x5
5!+ 21476
x6
6!+ 1551368
x7
7!+ · · · ,
P(x) = 1 + x + x2 + 2 x3 + 5 x4 + 16 x5 + 78 x6 + 588 x7 + 8047 x8 + 205914 x9 + · · · ,
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CHAPTER 4. POINT-DETERMINING GRAPHS
ZP = 1 + p1 +
(1
2p2
1 +1
2p2
)+
(1
3p3 + p1p2 +
2
3p3
1 +3
2p2
1p2
)
+
(1
2p4 +
4
3p4
1 + p22 +
2
3p1p3
)
+
(p2
1p3 +49
10p5
1 +11
3p3
1p2 +1
3p2p3 +
3
5p5 + p1p4 +
9
2p1p
22
)+ · · · .
Figure 4.3. Unlabeled point-determining graphs on n vertices, n ≤ 4.
In this way, the molecular decomposition of P takes the form
P = X + E2 +
(XE2 + E3
)+
(X2
E2 + XE3 + E2 E2 + X2E2 + E4
)+ · · · .
Let pn, cn and gn be numbers of labeled point-determining graphs, connected
graphs and graphs on n vertices. We have the following table:
4.2. Connected Point-Determining Graphs and Connected
Co-Point-Determining Graphs
A point-determining or a co-point-determining graph is called connected if the
graph itself is connected. We denote by Pc the species of connected point-determining
graphs, and by Qc the species of connected co-point-determining graphs.
91
CHAPTER 4. POINT-DETERMINING GRAPHS
Table 1. Numbers of labeled point-determining graphs, connectedgraphs and graphs on n vertices, n ≤ 10.
n pn cn gn
1 1 1 12 1 1 23 4 4 84 32 38 645 588 728 10246 21476 26704 327687 1551368 1866256 20971528 218608712 251548592 2684354569 60071657408 66296291072 68719476736
10 32307552561088 34496488594816 35184372088832
Theorem 4.2.1. Let P be the species of point-determining graphs, and let Pc be
the species of connected point-determining graphs. We have
P = (1 + X) E (Pc − X).
Proof. Since a point-determining graph can have at most one isolated vertex,
and the rest of its connected components are connected point-determining graphs
with at least two vertices, we have
P = E (Pc − X) + XE (Pc − X).
Theorem 4.2.2. For the species Q and Qc, we have
Q = E (Qc).
Proof. The result follows straightforwardly from the observation that a graph
is co-point-determining if and only if all its connected components are.
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CHAPTER 4. POINT-DETERMINING GRAPHS
Since P and Q are isomorphic to each other, we get
(4.2.1) P = (1 + X) E (Pc − X) = E (Qc).
Recall the virtual species Γ, which is the compositional inverse of E+ introduced
in Section 2.4, we have
Qc = Γ(P+),
where P+ is the species of point-determining graphs with nonempty vertex sets.
Hence we get to compute the associated series of Qc:
Qc(x) = log(P(x)),
Qc(x) =∑
k≥1
µ(k)
klog(P(xk)),
ZQc =∑
k≥1
µ(k)
klog(ZP pk − pk + 1).
We write down the first several terms of the associated series of Qc using Maple:
Qc(x) = x + 3
x3
3!+ 19
x4
4!+ 462
x5
5!+ 18268
x6
6!+ 1410394
x7
7!+ 206677954
x8
8!+ . . . ,
Qc(x) = x + x3 + 3x4 + 11x5 + 61x6 + 507x7 + 7442x8 + 197772x9 + 9808209x10 + · · · ,
ZQc = p1 +
(1
2p1p2 +
1
2p3
1
)+
(19
24p4
1 +3
4p2
1p2 +1
3p1p3 +
7
8p2
2 +1
4p4
)
+
(7
3p3
1p2 +1
6p2p3 +
77
20p5
1 +13
4p1p
22 +
1
2p2
1p3 +2
5p5 +
1
2p1p4
)+ · · · .
Equation (4.2.1) can be rewritten as
(4.2.2) P = E (Γ + Pc − X) = E (Qc).
93
CHAPTER 4. POINT-DETERMINING GRAPHS
As a general fact, if E (F1) = E (F2) for any species F1 and F2, then
E+(F1) = E+(F2).
It follows that Γ E+(F1) = Γ E+(F2). Since Γ E+ = X, we have F1 = F2.
Therefore, Equation (4.2.2) gives
Γ + Pc − X = Q
c,
or, equivalently,
Γ = Qc − P
c≥2,
which gives an explicit expression for the virtual species Γ as the difference of two
species.
We also get the functional equations relating the associated series of Pc and those
of Qc:
Pc(x) = Q
c(x) + x − log(1 + x),
Pc(x) = Qc(x) + x − (x − x2),
ZPc = ZQc + p1 −∑
k≥1
µ(k)
klog(1 + pk).
Through computation, we get the first several terms of the associated series of
Pc:
Pc(x) = x +
x2
2!+
x3
3!+ 25
x4
4!+ 438
x5
5!+ 18388
x6
6!+ 1409674
x7
7!+ 206682994
x8
8!+ . . . ,
Pc(x) = x + x2 + x3 + 3x4 + 11x5 + 61x6 + 507x7 + 7442x8 + 197772x9 + · · · ,
ZPc = p1 +
(1
2p2
1 +1
2p2
)+
(1
6p3
1 +1
3p3 +
1
2p1p2
)
94
CHAPTER 4. POINT-DETERMINING GRAPHS
+
(25
24p4
1 +5
8p2
2 +3
4p2
1p2 +1
3p1p3 +
1
4p4 +
1
2p2
1p3 +13
4p1p
22
)
+
(7
3p3
1p2 +3
5p5 +
73
20p5
1 +1
6p2p3 +
1
2p1p4
)+ · · · .
Let pcn and qc
n be the numbers of labeled connected point-determining graphs and
connected co-point-determining graphs, respectively, so that
Pc(x) =
∑
n≥1
pcn
xn
n!, Q
c(x) =∑
n≥1
qcn
xn
n!.
We get from the above species equivalence an identity relating pcn and qc
n:
(4.2.3) pcn + (−1)n−1(n − 1)! = qc
n, for n ≥ 2.
Figure 4.4 shows the unlabeled connected point-determining graphs and unlabeled
connected co-point-determining graphs with 4 vertices.
p1 p2 p3 q1 q2 q3
Figure 4.4. Unlabeled connected point-determining graphs and un-labeled connected co-point-determining graphs on 4 vertices.
We can get the number of labeled connected point-determining graphs and co-
point-determining graphs by calculating the number of labeled graphs isomorphic to
each pi and qi:
pc4 =
(24
| aut(p1)|+
24
| aut(p2)|+
24
| aut(p3)|
)= 12 + 12 + 1 = 25,
qc4 =
(24
| aut(p1)|+
24
| aut(p2)|+
24
| aut(p3)|
)= 12 + 3 + 4 = 19.
95
CHAPTER 4. POINT-DETERMINING GRAPHS
A combinatorial proof of Equation (4.2.3) would be desirable.
Table 2. Values of pcn and qc
n , for n ≤ 10.
n pcn qc
n
1 1 12 1 03 1 34 25 195 438 4626 18388 182687 1409674 14103948 206682994 2066779549 58152537184 58152577504
10 31715884061624 31715883698744
4.3. Bi-Point-Determining Graphs
Definition 4.3.1. A graph is called bi-point-determining if it is both point-determining
and co-point-determining.
The enumeration of the bi-point-determining graphs is carried out through the
structures called phylogenetic trees.
Definition 4.3.2. A phylogenetic tree is a rooted tree with labeled leaves and unla-
beled internal vertices in which no vertex has exactly one child.
24
5
1
3
6
Figure 4.5. A phylogenetic tree on [6].
96
CHAPTER 4. POINT-DETERMINING GRAPHS
Theorem 4.3.3. Let R be the species of bi-point-determining graphs. Let G and
S be the species of graphs and phylogenetic trees respectively. Then we have
G = R(2S − X).
The enumeration of phylogenetic trees was studied by Carlitz and Riordan [5],
Foulds and Robinson [7], Lomnicki [14], and others.
Lemma 4.3.4. Let S be the species of phylogenetic trees, and let E≥2 be the species
of sets with no less than two elements. We have
(4.3.1) S = X + E≥2(S ).
Proof. A phylogenetic tree is either a singleton vertex, which contributes to the
term X on the right-hand side of Equation (4.3.1), or is a phylogenetic tree with at
least two leaves, in which case we can separate the root and the rest of the tree and
get a set of at least two subtrees each of which is again a phylogenetic tree, which, as
illustrated in Figure 4.6, is an E≥2 S -structure.
Figure 4.6. A phylogenetic tree with at least two leaves is a set ofat least two subtrees each of which is a phylogenetic tree.
97
CHAPTER 4. POINT-DETERMINING GRAPHS
Equation (4.3.1) enables us to compute the first several terms of the associated
series of S using Maple:
S (x) = x +x2
2!+ 4
x3
3!+ 26
x4
4!+ 236
x5
5!+ 2752
x6
6!+ 39208
x7
7!+ 660032
x8
8!+ · · · ,
S (x) = x + x2 + 2x3 + 5x4 + 12x5 + 33x6 + 90x7 + 261x8 + 766x9 + · · · ,
ZS = p1 +
(1
2p2
1 +1
2p2
)+
(2
3p3
1 + p1p2 +1
3p3
)
+
(13
12p4
1 + p21p2 +
2
3p1p3 +
3
4p2
2 +1
2p4
)
+
(59
30p5
1 +4
3p2
1p3 + 2 +5
2p1p
22 +
13
3p3
1p2 + p1p4 +2
3p2p3 +
1
5p5
)
+
(172
45p6
1 +2
5p1p5 +
59
6p4
1p2 +26
9p3
1p3 +15
2p2
1p22 + 2p2
1p4 +8
3p1p2p3
+3
2p2p4 +
1
2p6
)+ · · · .
Figure 4.7 shows the unlabeled phylogenetic trees with no more than 5 vertices.
Figure 4.7. Unlabeled phylogenetic trees on n vertices, n ≤ 5.
98
CHAPTER 4. POINT-DETERMINING GRAPHS
Therefore, the molecular decomposition of S is
S = X + E2 +
(XE2 + E3
)+
(X2
E2 + E2 E2 + E22 + XE3 + E4
)
+
(X3
E2 + X2E3 + 3XE
22 + 2XE2 E2 + XE4 + 3E2E3 + E5
)+ · · · .
It is of interest to observe that different trees might have the same species ex-
pression. For example, the three unlabeled phylogenetic trees with 5 vertices listed
in Figure 4.8 correspond to the same species XE 22 .
Figure 4.8. Unlabeled phylogenetic trees corresponding to thespecies XE 2
2 .
Definition 4.3.5. We define an alternating phylogenetic tree to be either a single
vertex, or a phylogenetic tree with more than one labeled vertex whose internal ver-
tices are colored black or white, where no two adjacent vertices are colored the same
way.
We denote by S ∗ the species of alternating phylogenetic trees. The structure of
alternating phylogenetic trees S ∗ was studied by Stanley [27, p. 89], MacMahon [16,
17], and Riordan and Shannon [24] under other names such as yoke-chains and series-
parallel networks.
Note that any phylogenetic tree with more than one labeled vertex gives rise to two
alternating phylogenetic trees, since the coloring of the internal vertices is uniquely
99
CHAPTER 4. POINT-DETERMINING GRAPHS
8
3
7
69
5
1
2
4
Figure 4.9. An alternating phylogenetic tree labeled on the set [9],where the root is colored black.
determined by the coloring of the root, which has two choices. Therefore,
S∗ = 2S − X.
Proof of Theorem 4.3.3. In this proof, we will show how to transform an
arbitrary graph into a bi-point-determining graphs. This transformation, with all in-
formation preserved using the structure of alternating phylogenetic trees, is reversible,
and hence gives a bijection from the set of graphs with vertex set U to the set of triples
of the form (π, ϕ, γ) where π is a partition of U , ϕ is a bi-point-determining on the
blocks of π, and γ is a set of alternating phylogenetic trees each of which is labeled
on a block of π. Such a bijection leads to the functional equation
G = R S∗.
Let (g, h) be an ordered pair that satisfies the following conditions
C1 g is a graph with vertex set u = u1, u2, . . . , where each ui is a set, and the
ui are disjoint;
C2] h is a set h = t1, t2, . . ., where each ti is an alternating phylogenetic tree
labeled on the vertex set ui. Such pairs (g, h) are illustrated in Figures 4.10 and 4.11.
100
CHAPTER 4. POINT-DETERMINING GRAPHS
t3t2
g with V (g) = u1, u2, u3, u4
t4t1
h = t1, t2, t3, t4
u1
u4u2 u3
Figure 4.10. A pair (g, h) that satisfies conditions C1 and C2.
Next, we define two operations OQ and OP on pairs (g, h) such that each operation
sends (g, h) to a new ordered pair (g′, h′).
First, the operation OQ sends (g, h) to a new pair (g′, h′), where g′ is a graph and
h′ is a set of alternating phylogenetic trees. More precisely, we start by defining an
equivalence relation on the vertex set of g such that two vertices are equivalent if they
have the same augmented neighborhoods. We denote by
E = e1, e2, . . . , em
the set of equivalence classes of V (g), where each ei is a set of vertices of g, i.e.,
ei = ui,1, ui,2, . . . .
We let w1, w2, . . . be such that
wi = ui,1 ∪ ui,12 ∪ · · · .
We now let the wi be the vertices of g′, and two vertices wi and wj are adjacent in g′
if a vertex in the equivalence class ei is adjacent to a vertex in the equivalence class
101
CHAPTER 4. POINT-DETERMINING GRAPHS
ej in the graph g. Note that
w1 ∪ w2 ∪ · · · = u1 ∪ u2 ∪ · · · .
We construct a set
h′ = t′1, t′2, . . .
by letting t′i be the alternating phylogenetic tree whose root is colored white and whose
children are the trees ti,1, ti,2, · · · ∈ h labeled by the sets ui,1, ui,2, · · · ∈ V (g). Note that
t′i could fail to be an alternating phylogenetic tree, because some of the alternating
phylogenetic trees corresponding to the vertices ui,1, ui,2, . . . of g have white roots. In
this case, in order to make t′i have alternating colors on the internal vertices, we need
to modify our construction of t′i by taking each white-rooted alternating phylogenetic
tree, ti,k, and attaching the children of the root of ti,k directly to the root of t′i. We
get a new pair (g′, h′), where g′ is a graph on two vertices w1 and w2 with
w1 = u1, w2 = u2 ∪ u3 ∪ u4,
and h′ is a set of two alternating phylogenetic trees, one on the set w1, the other on
the set w2. Note that the alternating phylogenetic tree on u2 has a white root, so we
attach the children of this tree to the white root in the alternating phylogenetic tree
on w2.
Second, the operation OP sends the ordered pair (g, h) to a new ordered pair
(g′, h′) in a way similar to the operation OQ, except that now the equivalence classes
on the vertices of g are taken over vertices with the same neighborhoods in g, and
that the new root we attach to a set of alternating phylogenetic trees is colored black
instead of white. Again, we modify our construction as in OQ to ensure that we get
new alternating phylogenetic trees without violating the rule of alternating colors.
102
CHAPTER 4. POINT-DETERMINING GRAPHS
4 1
1
1
3
2
4
2
1
2,3
2 33 4
2 3
1 4
3
4
41 2
1
1
2
2,3,4
3
4
a)
(g′, h′)
(g, h) (g, h)
(g′, h′)
b)
Figure 4.11. a) An OQ application. b) An OP application.
Figure 4.12 illustrates the operation OP on a pair (g, h) in Figure 4.10, where
V (g) = u1, u2, u3, u4,
h is a set of alternating phylogenetic trees labeled on the set ui for i = 1, 2, 3, 4.
h′ = t′1, t′2
t′1
t′2
g′ with V (g′) = w1, w2
t′2
t′1
w1
w2
Figure 4.12. The operation OP applied to a pair (g, h) in Figure 4.10requires a modification on the construction of alternating phylogenetictree t′2.
103
CHAPTER 4. POINT-DETERMINING GRAPHS
It is straightforward to check that the new pair (g′, h′) we get under either the
application of OQ or OP satisfies conditions C1 and C2, and that the union of the
vertices of g is equal to the union of the vertices of g′. We further observe that the
operation OQ sends a graph g to a co-point-determining graph g′, since in g′, no two
vertices have the same augmented neighborhoods. Similarly, the operation OP sends
a graph g to a point-determining graph g′. That is why OP and OQ should be applied
alternatingly, since OP has no effect on a point-determining graph, and OQ has no
effect on a co-point-determining graph.
Now let G be a graph with vertex set
U = v1, v2, . . . .
We relabel G with the set
U ′ = U1, U2, . . . ,
where each Ui = vi is a singleton set, viewed as the label of a vertex of G. We
assign to G a set
H = T1, T2, . . . ,
where each Ti is a single vertex vi. Then (G, H) is an ordered pair satisfying conditions
C1 and C2.
We keep applying the operations OQ and OP alternatingly on the pair (G, H),
until neither operation has any effect, that is, when we reach an ordered pair (Gk, Hk)
in which Gk is a bi-point-determining graph. Illustrated in Figure 4.13 is a sequence
of alternating operations OQ and OP , starting with OP , sending a pair (G, H) to a
pair (Gk, Hk), where Gk is a bi-point-determining graph on a single vertex, and Hk is
a set consists of a single alternating phylogenetic tree whose vertex set is the vertex
of Gk.
104
CHAPTER 4. POINT-DETERMINING GRAPHS
2,3
5
1
4
4
1
2
2
1
3
54
4,5
1
1 2 3 4 5
3
1,2,3,4,5
4,5
2
5
1
4
3
3
3
2
2
321 4 52,3,4,5
1
1 2 3
1
4
1
5
2,3,4,5
5
Figure 4.13. Alternating applications of OP and OQ send a pair(G, H) to a pair (Gk, Hk).
The above discussion shows that we get from any graph G, we get a pair (Gk, Hk)
satisfies conditions C1 and C2. To be more precise, writing
V (Gk) = V1, V2, . . . , Hk = T1, T2, . . . ,
we get a triple of the form (π, ϕ, γ), where
i) π = V1, V2, . . . is a partition of U ;
ii) ϕ = Gk is a bi-point-determining graph with vertex set π.
iii) γ = T1, T2, . . . , where for each i, Ti is an alternating phylogenetic tree with
vertex set Vi.
On the other hand, since all information is preserved using the alternating phylo-
genetic trees, the above procedure is reversible. Roughly speaking, on each step, the
adjacency outside the equivalence classes are always preserved, while the adjacency
between vertices with the same augmented neighborhoods are transformed into them
105
CHAPTER 4. POINT-DETERMINING GRAPHS
located in the same alternating phylogenetic tree with a white common ancestor,
where the common ancestor of two vertices a and b in a phylogenetic tree is defined
to be such that if we take the unique shortest path from a to b, say, w0w1 · · ·wl, with
w0 = a and wl = b, then the common ancestor of a and b is the unique wi for which
both wi−1 and wi+1 are children of wi.
More precisely, suppose we are given a triple (π, ϕ, γ), where π = V1, V2, . . . is
a partition of U , i.e., ∪iVi = U , ϕ is a bi-point-determining graph on the blocks of π,
and γ is a set S1, S2, . . . in which each Si is an alternating phylogenetic tree labeled
on the set Vi. Then there is a unique graph G with vertex set U constructed in the
way such that for any v1, v2 ∈ V (G), v1, v2 is an edge of G if and only if exactly
one of the following two conditions is satisfied:
• v1, v2 ∈ Vi for some i, hence v1 and v2 are labels of vertices of Si. Then we
require the common ancestor of those vertices labeled v1 and v2 in Si to be
colored white. The common ancestor of two vertices a and b in a phylogenetic
tree is defined to be such that if we take the unique shortest path from a to
b, say, w0w1 · · ·wl, with w0 = a and wl = b, then the common ancestor of a
and b is the unique wi for which both wi−1 and wi+1 are children of wi.
• v1 ∈ Vi, v2 ∈ Vj and i 6= j. Then we require Vi and Vj to be adjacent to each
other in the graph ϕ.
The above can be concluded with a species equivalence
G = R S∗,
which gives
S = R (2S − X).
106
CHAPTER 4. POINT-DETERMINING GRAPHS
86
2
4
1
8
6 3
3
4
7
1 527
5
V1 V2
V4V3
V2
V3 V4
V1
Figure 4.14. Construct a graph G from a given triple (π, ϕ, γ).
We get from Theorem 4.3.3 a functional equation expressing the species R in
terms of the species G .
Corollary 4.3.6. Let G , R be the species of graphs and the species of bi-point-
determining graphs. Let Γ be the virtual species as defined in Section 2.4. Then we
have
(4.3.2) R = G (2Γ − X).
Proof. First, we observe that since S0 = 0 and S1 = X, the same is true for
S ∗: S ∗0 = 0 and S ∗
1 = X. Proposition 19 of [2, p. 130] says that there exists a
unique virtual species S ∗〈−1〉 such that
S∗ S
∗〈−1〉 = S∗〈−1〉 S
∗ = X.
Since Theorem 4.3.3 gives that
R = G (S ∗〈−1〉),
107
CHAPTER 4. POINT-DETERMINING GRAPHS
it suffices to show that the compositional inverse of S ∗ is 2Γ − X.
But 2S − S ∗ = X implies that
S =X + S ∗
2.
On the other hand, Equation (4.3.1) implies 2S − X = E (S ) − 1. So
S∗ = E (S ) − 1,
and
(4.3.3) S∗ + 1 = E (S ) = E
(X + S ∗
2
).
But from
E Γ(X) = X + 1,
we get
E Γ S∗ = X S
∗ + 1,
so
E (Γ(S ∗)) = S∗ + 1 = E
(X + S ∗
2
)
by Equation (4.3.3). So
Γ(S ∗) =X + S ∗
2,
which gives that
X = 2Γ(S ∗) − S∗ = (2Γ − X) S
∗.
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CHAPTER 4. POINT-DETERMINING GRAPHS
Equation (4.3.2) gives rise to several identities for computing the associated series
of R:
R(x) = G (2 log(1 + x) − x),
R(x) = ZG (x − 2x2, x2 − 2x4, . . . ),
ZR = ZG
(2∑
k≥1
µk
klog(1 + pk) − p1, 2
∑
k≥1
µk
klog(1 + p2k) − p2, . . .
).
Using Maple, we get the following:
R(x) =x
1!+ 12
x4
4!+ 312
x5
5!+ 13824
x6
6!+ 1147488
x7
7!+ 178672128
x8
8!+ · · · ,
R(x) = x + x4 + 6x5 + 36x6 + 324x7 + 5280x8 + 156088x9 + 8415760x10 + · · · ,
(4.3.4)
ZR = p1 +
(1
2p4
1 +1
2p2
2
)+
(13
5p5
1 + 3p1p22 +
2
5p5
)
+
(96
5p6
1 + 11p21p
22 +
4
5p1p5 +
11
3p3
2 + p23 +
1
3p6
)+ · · · .
It is interesting to observe from Equation (4.3.4) that there is no bi-point-determining
graphs on 2 or 3 vertices. The unlabeled bi-point-determining graphs on 4 or 5 vertices
are shown in Figure 4.15.
Figure 4.15. Unlabeled bi-point-determining graphs on n vertices,n = 4, 5.
109
CHAPTER 4. POINT-DETERMINING GRAPHS
Therefore, we get the molecular decomposition of R:
R = X + E2 X2 + [5X(E2 X2) + D5] + · · · ,
where D5 = X5/D5 is the molecular species of regular pentagons with cycle index
ZD5 = Z(D5) =1
10(p5
1 + 5p1p22 + 4p5).
Listed in Table 3 are numbers of labeled and unlabeled bi-point-determining
graphs, denoted by rln and ru
n.
Table 3. Numbers of labeled and unlabeled bi-point-determininggraphs with no more than 15 vertices.
n rln ru
n
1 1 12 0 03 0 04 12 15 312 66 13824 367 1147488 3248 178672128 52809 52666091712 156088
10 29715982846848 841576011 32452221242518272 82079360012 69259424722321036032 14506388148013 291060255757818125657088 4679331014916814 2421848956937579216663491584 2779080872680384015 40050322614433939228627991906304 30630967638347575456
We denote by Rc the species of connected bi-point-determining graphs.
Theorem 4.3.7. For species R, Rc, and E+, we have
R = X + (1 + X) E+(Rc − X).
110
CHAPTER 4. POINT-DETERMINING GRAPHS
Proof. Since a graph with a singleton vertex is an Rc-structure, we denote by
Rc − X the species of connected bi-point-determining graphs with more than one
vertex.
Let R be a bi-point-determining graph. Then R is both point-determining and
co-point-determining. In particular, the decomposition of R into a set of its con-
nected components is similar with that of a point-determining graph in that there
is at most one connected component consisting of a singleton vertex. Hence such a
decomposition can result in one of the following three situations:
i) R is a singleton vertex, in which case we get an X-structure.
ii) R has more than one vertex, and none of its connected components is a singleton
vertex, in which case we get an E+(Rc − X).
iii) R has more than one vertex, and one of its connected components is a singleton
vertex, in which case we get an X · E+(Rc − X) species.
Therefore, we get the desired result.
We can compute the beginning terms of the associated series of Rc using Maple:
Rc(x) =
x
1!+ 12
x4
4!+ 252
x5
5!+ 12312
x6
6!+ 1061304
x7
7!+ 170176656
x8
8!
+ 51134075424x9
9!+ · · ·
Rc(x) = x + x4 + 5x5 + 31x6 + 293x7 + 4986x8 + · · · ,
ZRc = p1 +
(1
2p4
1 +1
2p2
2
)+
(21
10p5
1 +5
2p1p
22 +
2
5p5
)
+
(17
10p6
1 +17
2p2
1p22 +
2
5p1p5 +
11
3p3
2 + p23
1
3p6
)+ · · · .
The calculation agrees with the observation from Figure 4.15 that the only un-
labeled bi-point-determining graph on four vertices is connected, and among the six
111
CHAPTER 4. POINT-DETERMINING GRAPHS
unlabeled bi-point-determining graph on five vertices there is only one of them that
is not connected.
4.4. 2-Colored Graphs and Connected 2-colored Graphs
Definition 4.4.1. A proper coloring of a graph is an assignment of colors to the
vertices of the graph where no two adjacent vertices are assigned the same color. A
2-colored graph, also called a bi-colored graph by Harary [11, p. 93] and Hanlon [9],
is a graph in which all vertices are properly 2-colored.
For simplicity, we call the two colors in a 2-colored graph white and black. We
denote by G (X, Y ) the 2-sort species defined such that for a two-set U = (V, W ),
G [U ] is a 2-colored graph in which the vertices colored white are elements of V and
the vertices colored black are elements of W .
G (X, Y )X
Y
Y
X
Y
Figure 4.16. The 2-sort species of 2-colored graphs G (X, Y ).
Proposition 4.4.2. Let G (X, Y ) be the 2-sort species of 2-colored graphs. Then
the exponential generating series of G (X, Y ) is given by
G (x, y) =∞∑
m,n=0
2mn xm
m!
yn
n!.
112
CHAPTER 4. POINT-DETERMINING GRAPHS
Proof. In a 2-colored graph, each edge must connect one vertex of sort X and
one vertex of sort Y . Hence there are 2mn labeled 2-colored graphs with m white
vertices and n black vertices.
Theorem 4.4.3. Let G (X, Y ) be the 2-sort species of 2-colored graphs. Then the
cycle index of G (X, Y ) is given by
ZG (X,Y ) =∑
m,n≥0
( ∑
λ⊢m, µ⊢n
2∑
i,j gcd(λi, µj)pλ[x]
zλ
pµ[y]
zµ
).
Proof. Let (λ, µ) be an ordered pair of partitions, and let (σ, π) be an ordered
pair of permutations with σ having cycle type λ and π having cycle type µ. Let
fix (σ, π) = fix G [λ, µ] be the number of 2-colored graphs fixed by (σ, π).
To start with, we consider the simpler case when σ is a k-cycle and π is an l-cycle.
Let Kk,l denote the complete bipartite graph on [k, l], and let E(Kk,l) be its edge set.
Then |E(Kk,l)| = kl. Without loss of generality, we let the labeling of left-hand side
vertices of Kk,l be 1, 2, . . . , k, and the labeling of right-hand side vertices of Kk,l be
1′, 2′, . . . , l′. Then each edge of Kk,l is represented by an ordered pair (i, j′), for some
i ∈ [k] and j ∈ [l]. The pair of permutations (σ, π) acts on the set E(Kk,l) by letting σ
act on the set 1, 2, . . . , k and π act on the set 1′, 2′, . . . , l′. This action partitions
the kl edges of Kk,l into orbits A1, A2, . . . . We observe that there are lcm(k, l)
edges in each of the orbits, since all edges of the form (ir, j′r), where ir = σr(i) and
jr = πr(j) for some r = 1, 2 . . . , lcm(k, l)− 1, are in the same orbit as the edge (i, j ′),
and hence this action of (σ, π) on the set E(Kk,l) results in (kl)/lcm(k, l) = gcd(k, l)
orbits. Note that each 2-colored graph with vertex set [k, l] can be identified with a
subset of E(Kk,l). If a subset S of E(Kk,l) is fixed by the pair of permutations (σ, π),
then whenever an edge (i, j′) is in S, all edges in the same orbit as (i, j′) under the
action of (σ, π) on E(Kk,l) is in S as well. This means that the number of 2-colored
113
CHAPTER 4. POINT-DETERMINING GRAPHS
graphs fixed by the pair of permutations (σ, π) is the same as the number of subsets
of A1, A2, . . . , Agcd(k,l). Therefore,
fix (σ, π) = 2gcd(k,l).
For the general case, we write λ = (λ1, λ2, . . . ) and µ = (µ1, µ2, . . . ). It is straight-
forward to see that each ordered pair (λi, µj), for some integers i and j, gives rise to
a factor 2gcd λi,µj in the number fix (σ, π), and hence
fix G [λ, µ] =∏
i,j
2 gcd(λi, µj) = 2∑
i,j gcd(λi, µj).
Remark 4.4.4. The above argument also gives a way to count 2-colored graphs by
the number of edges. Let bm,n(x) be the ordinary generating function for 2-colored
graphs, in which m vertices are colored white and n vertices colored black, by the
number number of edges. We get the following expression for bm,n(x), which agrees
with the result of Harary and Palmer [11, p. 95]:
bm,n(x) =∑
λ⊢m, µ⊢n
1
zλzµ
m,n∏
k,l=1
(1 + xlcm(k, l)
)ck(λ)cl(µ) gcd(k, l)
,
where ci(λ) denotes the number of parts in λ with length i. For example, the coeffi-
cient of x4 in b2,3(x) is 3, as shown in Figure 4.17.
Figure 4.17. There are 3 unlabeled 2-colored graphs with 4 edgesand 5 vertices, 2 colored white, 3 colored black.
114
CHAPTER 4. POINT-DETERMINING GRAPHS
Theorem 4.4.3 enables us to calculate the associated series of G [X, Y ]:
G (x, y) = 1 +x
1!+
x2
2!+
x3
3!+
y
1!+
y2
2!+
y3
3!+ 2
x
1!
y
1!+ 4
x2
2!
y
1!+ 4
x
1!
y2
2!+ · · · ,
G (x, y) = 1 + x + y + 2xy + x2 + y2 + 3x2y + 3xy2 + x3 + y3 + · · · ,
ZG (X,Y ) = 1 + (p1[x] + p1[y]) +
(1
2p2
1[x] +1
2p2[x] + 2p1[x]p1[y] +
1
2p2[y] +
1
2p2
1[y]
)
+
(1
6p3
1[x] +1
2p1[x]p2[x] +
1
3p3[x] + p2[x]p1[y] + 2p2
1[x]p1[y]
+2p1[x]p21[y] + p1[x]p2[y] + +
1
3p3[y] +
1
2p1[y]p2[y] +
1
6p3
1[y]
)+ · · · .
Figure 4.18 shows the unlabeled 2-colored graphs on at most 4 vertices, where
each vertex is either a white vertex, a black vertex, or a vertex that can be colored
in both colors. For example, there are two unlabeled 2-colored graph on one vertex,
one color black the other colored white.
Figure 4.18. Unlabeled 2-colored graphs with no more than 4 vertices.
A 2-colored graph is called connected if the underlying graph is connected.
115
CHAPTER 4. POINT-DETERMINING GRAPHS
Theorem 4.4.5. Let G c(X, Y ) be the species of connected 2-colored graphs. Then
(4.4.1) G (X, Y ) = E (G c(X, Y )).
Proof. The canonical decomposition of a graph into connected components ap-
plies to 2-colored graphs.
Equation (4.4.1) is equivalent to
Gc(X, Y ) = Γ(G+(X, Y )),
which enables us to compute the associated series of G c[X, Y ] using Maple:
Gc(x, y) =
x
1!+
y
1!+
x
1!
y
1!+
x2
2!
y
1!+
x
1!
y2
2!+
x3
3!
y
1!+ 5
x2
2!
y2
2!+
x
1!
y3
3!+ · · · ,
G c(x, y) = x + y + xy + xy2 + x2y + x3y + xy3 + 2x2y2 + x4y + 4x3y2 + 4x2y3
+ xy4 + · · · ,
ZG c(X,Y ) = (p1[x] + p1[y]) + p1[x]p1[y] +
(1
2p2
1[x]p1[y] +1
2p1[x]p2
1[y] +1
2p2[x]p1[y]
+1
2p1[x]p2[y]
)+ · · · .
Figure 4.19. Unlabeled connected 2-colored graphs with no morethan 4 vertices.
116
CHAPTER 4. POINT-DETERMINING GRAPHS
4.5. Point-Determining 2-Colored Graphs
Definition 4.5.1. A 2-colored graph is called point-determining if the underlying
graph is point-determining. A 2-colored graph is called semi-point-determining if all
vertices of the same color have distinct neighborhoods.
Note that the notion of co-point-determining 2-colored graphs is not interesting,
since any two adjacent vertices in a 2-colored graph are colored differently, so that
there is no vertex that could be adjacent to both of them.
We denote by P(X, Y ) the species of point-determining 2-colored graphs, by
Ps(X, Y ) the species of semi-point-determining 2-colored graphs, and by Pc(X, Y )
the species of connected point-determining 2-colored graphs.
Theorem 4.5.2. For species P(X, Y ), Ps(X, Y ) and Pc(X, Y ), we have
Ps(X, Y ) = (1 + X)(1 + Y ) E (Pc
≥2(X, Y )),(4.5.1)
P(X, Y ) = (1 + X + Y ) E (Pc≥2(X, Y )).(4.5.2)
Proof. Let G1 be a semi-point-determining 2-colored graph. We observe that
a connected component of G1 could be either a single vertex colored white, a single
vertex colored black, or a connected point-determining 2-colored graph with at least
two vertices. At the same time, G1 can have at most one isolated vertex colored
with each color, due to the fact that all vertices in G1 of the same color must have
distinct neighborhoods. Equation (4.5.1) follows by translating the above into species
equivalence.
Let G2 be a point-determining 2-colored graph. As in the above discussion we see
that a connected component of G2 could be either a single vertex colored white, a
single vertex colored black, or a connected point-determining 2-colored graph with at
117
CHAPTER 4. POINT-DETERMINING GRAPHS
Pc≥2
(X, Y )
EPs(X, Y )
1 + Y
1 + X
Pc≥2
(X, Y )Pc≥2
(X, Y )
Pc≥2
(X, Y )
Figure 4.20. Ps(X, Y ) = (1 + X)(1 + Y ) E (Pc≥2(X, Y )).
least two vertices. But this time, since the underlying graph of G2 is point-determining
graph, G2 can have at most one isolated vertex in all. Hence the term (1 +X)(1 +Y )
in (4.5.1) is replaced with the term 1 + X + Y in (4.5.2).
1 + X + Y
Pc≥2
(X, Y )
Pc≥2
(X, Y )
Pc≥2
(X, Y )
EP(X, Y )
Pc≥2
(X, Y )
Figure 4.21. P(X, Y ) = (1 + X + Y ) E (Pc≥2(X, Y )).
Theorem 4.5.3. For the species G (X, Y ) and Ps(X, Y ), we have
G (X, Y ) = Ps(E+(X), E+(Y )).
Proof. The proof use the same idea as the proof of Theorem 4.1.3. To be more
precise, given any 2-colored graph, we define equivalence relations on the vertex set
by setting two same-colored vertices to be equivalent if they have the same neigh-
borhoods, and get a new 2-colored graph whose vertex set is the set of equivalence
118
CHAPTER 4. POINT-DETERMINING GRAPHS
classes and the adjacency in the original graph is accordingly preserved. We observe
that the resulting new graph is a semi-point-determining 2-colored graph, and the
rest is straightforward.
We make use of the virtual species Γ defined in Section 2.4 again, and get from
Theorem 4.5.3 that
Ps(X, Y ) = G (Γ(X), Γ(Y )),
which, together with Equations 4.5.1 and 4.5.2, allows us to compute the associ-
ated series of the species of semi-point-determining 2-colored graphs, the species of
point-determining 2-colored graphs, and the species of connected point-determining
2-colored graphs as follows:
Ps(x, y) = 1 +
x
1!+
y
1!+ 2
x
1!
y
1!+ 2
x
1!
y2
2!+ 2
x2
2!
y
1!+ 10
x2
2!
y2
2!+ 24
x2
2!
y3
3!
+ 24x3
3!
y2
2!+ · · · ,
Ps(x, y) = 1 + x + y + 2xy + x2y + xy2 + 3x2y2 + 3x3y2 + 3x2y3 + · · · ,
ZPs(X,Y ) = 1 + (p1[x] + p1[y]) + (2p1[x]p1[y]) + (p21[x]p1[y] + p1[x]p2
1[y])
+
(1
2p2[x]p2[y] +
5
2p2
1[x]p21[y]
)
+ (p1p2[x]p2[y] + p2[x]p1p2[y] + 2p31[x]p2
1[y] + 2p21[x]p3
1[y]) + · · · .
P(x, y) = 1 +x
1!+
y
1!+
x
1!
y
1!+ 2
x
1!
y2
2!+ 2
x2
2!
y
1!+ 6
x2
2!
y2
2!+ 24
x2
2!
y3
3!
+ 24x3
3!
y2
2!+ · · · ,
P(x, y) = 1 + x + y + xy + x2y + xy2 + 2x2y2 + 3x3y2 + 3x2y3 + · · · ,
ZP(X,Y ) = 1 + (p1[x] + p1[y]) + (p1[x]p1[y]) + (p21[x]p1[y] + p1[x]p2
1[y])
+
(1
2p2[x]p2[y] +
3
2p2
1[x]p21[y]
)
119
CHAPTER 4. POINT-DETERMINING GRAPHS
+ (p1p2[x]p2[y] + p2[x]p1p2[y] + 2p31[x]p2
1[y] + 2p21[x]p3
1[y]) + · · · .
Pc(x, y) =
x
1!+
y
1!+
x
1!
y
1!+ 4
x2
2!
y2
2!+ 6
x2
2!
y3
3!+ 6
x3
3!
y2
2!+ · · · ,
Pc(x, y) = x + y + xy + x2y2 + x3y2 + x2y3 + · · · ,
ZPc(X,Y ) = (p1[x] + p1[y]) + (p1[x]p1[y]) + (p21[x]p2
1[y])
+
(1
2p1p2[x]p2[y] +
1
2p2[x]p1p2[y] +
1
2p3
1[x]p21[y] +
1
2p2
1[x]p31[y]
)+ · · · .
Figure 4.22. Unlabeled point-determining 2-colored graphs with nomore than 5 vertices.
We can write down the molecular decomposition of a 2-sort species. For example,
Figure 4.22 gives the first terms of the molecular decomposition of P(X, Y ).
P(X, Y ) = 1 + (X + Y ) + XY + (X + Y )(XY ) + (E2(X)E2(Y ) + X2Y 2)
+[2(X + Y )E2(XY ) + X3Y 2 + X2Y 3
]+ · · · ,
where the terms 2(X +Y )E2(XY )+X3Y 2 +X2Y 3 correspond to the unlabeled point-
determining 2-colored graphs with 5 vertices as shown in Figure 4.23.
120
CHAPTER 4. POINT-DETERMINING GRAPHS
XE2(XY )Y E2(XY )
X2Y 3
XE2(XY )
X3Y 2 Y E2(XY )
Figure 4.23. Unlabeled point-determining 2-colored graphs with 5 vertices.
121
APPENDIX A
Index of Species
0 empty species.
1 singleton set species.
X species of singletons.
Xn species of linear orders of order n.
Xn/A molecular species of (left) cosets of A in Sn.
A (X, Y ) 2-sort species of rooted trees.
C (X) species of oriented cycles.
Dn(X) molecular species Xn/Dn.
E (X) species of sets.
Ek(X) species of k-element sets.
Ek〈F 〉 exponential composition of species F of order k, for k ≥ 1.
E 〈F 〉 exponential composition of species F.
K (X) species of complete graphs.
G (X) species of (simple) graphs.
G c(X) species of connected graphs.
G (X, Y ) 2-sort species of 2-colored graphs, or bi-colored graphs.
G c(X, Y ) 2-sort species of connected 2-colored graphs.
122
CHAPTER A. INDEX OF SPECIES
Γ(X) virtual species of “connected (1 + X)-structures”, or
compositional inverse of E+.
L (X) species of linear orders.
N (X) species of (2-)rectangles.
N (k)(X) species of k-rectangles.
NN species of k-dimensional cubes on [N ], or E ⊡kn [N ], where N = nk.
OG(X) species associated to a graph G.
P(X) a) species of point-determining graphs (in chapter 4);
b) species of prime graphs.
Pc(X) species of connected point-determining graphs.
P(X, Y ) 2-sort species of point-determining 2-colored graphs.
Ps(X, Y ) 2-sort species of semi-point-determining 2-colored graphs.
Pc(X, Y ) 2-sort species of connected point-determining 2-colored graphs.
Φ(X) species of permutations.
Ψ(k)(X) species of partitions into k blocks .
Q(X) species of co-point-determining graphs.
Qc(X) species of connected co-point-determining graphs.
R(X) species of bi-point-determining graphs.
Rc(X) species of connected bi-point-determining graphs.
S (X) species of phylogenetic trees.
S ∗(X) species of alternating phylogenetic trees, or yoke-chains, or
series-parallel networks.
123
APPENDIX B
General Notations
A > B product of groups A and B acting on the set [m + n].
A × B product of groups A and B acting on the set [mn].
B ≀ A wreath product of groups A and B.
BA exponentiation group of A and B.
aut(G) automorphism group of a graph G.
B category of finite sets with bijections.
Bk category of k-sets with bijective multifunctions.
bn number of unlabeled prime graphs of order n.
C set of unlabeled connected graphs.
CM monoid algebra associated with a free commutative monoid M.
ck(λ) number of parts of length k in a partition λ.
cn number of labeled connected graphs of order n.
cn number of unlabeled connected graphs of order n.
c. t.(σ) cycle type of a permutation σ.
dλ number of permutations with cycle type λ.
Dn dihedral group of order n.
D(F ) Dirichlet exponential generating function of a species F.
D(G) Dirichlet exponential generating function of a graph G.
124
CHAPTER B. GENERAL NOTATIONS
E(G) edge set of a graph G.
gcd(k, l) greatest common divisor of integers k and l.
f1 ∗ f2 image of f2 under the operator obtained by substituting the
operator Ir for the variables pr in f1.
F (x) exponential generating series of a species F.
F (x) type generating series of a species F.
F1 + F2 sum of species F1 and F2.
F1 · F2 product of species F1 and F2.
F1 F2 composition of species F1 and F2.
F1 ⊡ F2 arithmetic product of species F1 and F2.
G1 ⊙ G2 Cartesian product of graphs G1 and G2.
λi ith part of the partition λ = (λ1, λ2, . . . ), arranged in
weakly decreasing order.
Kk,l complete bipartite graph on the set [k, l].
l(G) number of vertices in a graph G.
L(G) number of graphs isomorphic to G with vertex set V (G).
lcm(k, l) least common multiple of integers k and l.
N set of natural numbers.
Ω Q-algebra generated by the operators Ik.
P set of prime numbers.
P set of unlabeled prime graphs with respect to the Cartesian
multiplication.
125
CHAPTER B. GENERAL NOTATIONS
pn power sum symmetric function of order n.
pλ power sum symmetric function indexed by the partition λ.
pλ ⊠ pµ an operation defined by Definition 1.2.3.
Parn set of partitions of n.
Parkn set of sequences of k partitions of n.
Q set of rational numbers.
R ring of polynomials in the variables p1, p2, . . . with the operation ⊡ .
Sn symmetric group of order n.
V (G) vertex set of a graph G.
zλ number of permutations commute with a permutation of cycle type λ.
Z(A) Polya’s cycle index polynomial of a permutation group A.
ZF cycle index of F.
Z(G) = ZOGcycle index of the species associated to the graph G.
126
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