Arithmetic Product of Species

Cartesian Product of Graphs
and Arithmetic Product of Species
MIT Combinatorics Seminar

Ji Li

Department of Mathematics
Brandeis University

March 9th, 2007

Cartesian Product
Species
Arithmetic Product of Species
Exponential Composition of Species

What This Talk Is About

Graphs Species

Arithmetic
Cartesian Product of
Product of Species of Species
Graphs Graphs

Exponential
Composition
of Species

Prime Graphs

J. L. Cartesian Product of Graphs and Arithmetic Product of Species

Cartesian Product
Species Definition of Cartesian Product
Arithmetic Product of Species Unique Factorization of Connected Graphs

Outline
Definition of Cartesian Product
Unique Factorization of Connected Graphs
Species
Definition of Species
Species of Graphs
Species Associated to A Graph
Rectangles
Cartesian Product and Arithmetic Product
Enumerating Prime Graphs

Cartesian Product


Deﬁnition
The Cartesian product of two graphs G1 and G2 , denoted G1 ⊡ G2 ,
is the graph whose vertex set is

V (G1 ⊡ G2 ) = {(u, v ) : u ∈ V (G1 ), v ∈ V (G2 )},

in which (u, v ) is adjacent to the vertex (w , z) if either
u = w and {v , z} ∈ E (G2 )
or
v = z and {u, w } ∈ E (G1 ).


Cartesian Product

An Example

Example

1, 1′ 1, 2′

1, 3′ 1, 4′
1 1′ 2′

=
2 3 3′ 4′
2, 1′ 2, 2′ 3, 1′ 3, 2′

2, 3′ 2, 4′ 3, 3′ 3, 4′

The Cartesian product of a graph on 3 vertices and a graph on 4
vertices is a graph on 12 vertices.

Cartesian Product

Properties

◮ commutativity:
G1 ⊡ G2 ∼ G2 ⊡ G1
=
◮ associativity:

(G1 ⊡ G2 ) ⊡ G3 ∼ G1 ⊡ (G2 ⊡ G3 )
=

◮ · We write
⊡ Gi = G1 ⊡ G2 ⊡ · · · ,
i ∈I
n
Gn = ⊡ G.
i =1


Cartesian Product

Prime Graph

Deﬁnition
A graph G is said to be prime with respect to Cartesian
multiplication if G satisﬁes all of the following conditions:

a) G is not a singleton vertex.
b) G is connected.
c) G ∼ H1 ⊡ H2 implies that H1 or H2 is a singleton vertex.
=


Cartesian Product

Relatively Prime

Deﬁnition
Two graphs G and H are called relatively prime with respect to
Cartesian multiplication if and only if
G = G1 ⊡ J and H ∼ H1 ⊡ J
∼ =
imply
J is a singleton vertex.


Cartesian Product

Decomposition of a Connected Graph

◮ · If G is a connected graph, then G can be decomposed into
prime factors. That is, there is a set {Pi }i ∈I of prime graphs such
that
G ∼ ⊡ Pi .
=
i ∈I

◮ · Sabidussi proved that such a prime factorization is unique up
to isomorphism.


Cartesian Product

An Example

Example

= ⊡ ⊡

A connected graph on 12 vertices is decomposed into two prime
graphs on 2 vertices and one prime graph on 3 vertices.


Cartesian Product

Monoid Structure of Unlabeled Connected Graphs

◮ · Let M be the set of unlabeled connected graphs. Let P be the
set of unlabeled prime graphs.
The unique factorization theorem of Sabidussi gives
M the structure of a commutative free monoid with a
set of primes P.
◮ · This is saying —
Every element of M has a unique factorization of the
e e e
form b11 b22 · · · bkk , where the bi are distinct primes in P.


Cartesian Product

An Equation Relating M and P

◮ · Let l (G ), the number of vertices in G , be a length function for
M. Then we get an equation relating M and P:

1 1
= .
l (G )s 1 − l (P)−s
G ∈M P∈P


Cartesian Product

Number of Unlabeled Prime Graphs

Theorem
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
cn 1
= .
ns (1 − m−s )bm
n≥1 m≥2


Cartesian Product

A Table

◮ · cn is the number of unlabeled connected graphs on n vertices
◮ · bn is the number of unlabeled prime graphs on n vertices

n 1 2 3 4 5 6 7 8
cn 1 1 2 6 21 112 853 11117
bn 1 1 2 5 21 110 853 11111


Cartesian Product

Unlabeled Prime Graphs on 4 Vertices
Example

There are 5 unlabeled prime graphs on 4 vertices.
◮ · In fact, there is only one unlabeled connected graph on 4
vertices that is not prime:


Cartesian Product
Species
Species of Graphs

Outline
Species
Species of Graphs
Rectangles

Cartesian Product
Species
Species of Graphs

Species
Definition
A species of structures is a functor from the category of finite sets,
with bijections for morphisms, to itself.
Example
We denote by G the species of graphs. This means the following:
◮ for any finite set U, G [U] is the set of graphs with vertex set
U;
◮ any bijection
σ:U→V
will induce a bijection

G [σ] : G [U] → G [V ],

Cartesian Product
Species
Species of Graphs

Examples

Example
◮ The species of singletons X is defined by setting

{U}, if |U| = 1,
X [U] =
∅, otherwise.

◮ The species of sets E is defined by setting E [U] = {U}. In
other words, the set of E -structures on a given finite set U is
a singleton set.
◮ The species of linear orders L . In particular, the species of
linear orders on n-element sets is denoted by X n .
◮ The species of connected graphs C .


Cartesian Product
Species
Species of Graphs

Addition

We can deﬁne operations on species, such as

= or

F1 F2
F1 + F2

Addition: F1 + F2 .


Cartesian Product
Species
Species of Graphs

Multiplication

=
F1 F2
F1 · F2

Multiplication: F1 · F2 .


Cartesian Product
Species
Species of Graphs

Composition

F2
F2
= F2 = F2
F1 F2 F1
F1 ◦ F2 F2

Substitution: F1 ◦ F2 = F1 (F2 ).


Cartesian Product
Species
Species of Graphs

Associated Series

Each species is associated with three counting series—
◮ The exponential generating series F (x) counts labeled
F -structures;
◮ The type generating series F (x) counts unlabeled
F -structures;
◮ The cycle index ZF is a symmetric function in the variables
p1 , p2 , . . . which satisﬁes

F (x) = ZF (x, 0, 0, . . . ),
F (x) = ZF (x, x 2 , x 3 , . . . ).


Cartesian Product
Species
Species of Graphs

Cycle Index of G
We can calculate the cycle index of the species G using a formula
given by H. D´coste, G. Labelle and P. Leroux:
e

2 4 3 2
ZG = p1 + (p1 + p2 ) + p + 2p1 p2 + p3 +
3 1 3
8 4 2 2 4
p1 + 4p1 p2 + 2p2 + p1 p3 + p4 +
3 3
128 5 32 3 2 8 2 4 4
p + p p2 + 8p1 p2 + p1 p3 + p2 p3 + 2p1 p4 + p5
15 1 3 1 3 3 5
+ ··· ,


Cartesian Product
Species
Species of Graphs

Connected Graphs

1 2 8

3
4 5 7 6

The fact that a graph is an assembly of its connected components
gives rise to a species identity

G = E ◦C,


Cartesian Product
Species
Species of Graphs

Cycle Index of C

...that can be used to compute the cycle index of connected graphs

1 2 1 1 2 3
ZC = p1 + p + p2 +
p3 + p1 + p1 p2 +
2 1 2 3 3
19 4 2 5 2 2 1
p1 + 2p1 p2 + p2 + p1 p3 + p4 +
12 4 3 2
19 3 2 91 5 2 4 2 3
p1 p2 + p2 p3 + p1 + 5p1 p2 + p1 p3 + p5 + p1 p4
3 3 15 3 5
+ ··· .


Cartesian Product
Species
Species of Graphs

Definition

Definition
For any graph G , there is a species associated to G , denoted OG .
The OG -structures on a finite set U is the set of graphs isomorphic
to G with vertex set U.

Example
The species E2 is the species of sets with 2 elements. It is also the
species associated to the graph


Cartesian Product
Species Rectangles
Arithmetic Product of Species Arithmetic Product of Species

Outline
Species
Species of Graphs
Rectangles

Cartesian Product
Species Rectangles

Definition

The following definition was given by Maia and Mendez (2006).
Definition
Let m and n be integers.
A rectangle on the set [mn] of height m is a pair (π1 , π2 ) such that
a) π1 is a partition of [mn] with m blocks, each of size n
b) π2 is a partition of [mn] with n blocks, each of size m
c) if B is a block of π1 and B ′ is a block of π2 then |B ∩ B ′ | = 1.

◮ · We denote by N the species of rectangles.


Cartesian Product
Species Rectangles

A Rectangle
Example

12 7 11 6 1 3 4 9

8 10 2 5 ∼ 8 5 2 10

1 9 4 3 12 6 11 7

These two pictures represent the same rectangle.
◮ · This is a rectangle on the set [12] = {1, 2, . . . , 12} of height 3:

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}} .


Cartesian Product
Species Rectangles

An Equivalent Picture

Example
1 1
A1 9 8
B1
4 12
3
12 7 11 6
8 9
A2 10
2
10
7
B2
5
8 10 2 5 =
12 4
7 2
A3 6
11 11 B3
1 9 4 3 5 3
π1 = {A1 , A2 , A3 } 6
B4
π2 = {B1 , B2 , B3 , B4 }


Cartesian Product
Species Rectangles

Deﬁnition of k-Rectangle

Deﬁnition
Let m1 , m2 , . . . , mk be integers. Let n = m1 m2 · · · mk . A
k-rectangle on the set [n] is a k-tuple of partitions (π1 , π2 , . . . , πk )
such that
a) for each i = 1, 2, . . . , k, πi has mi blocks, each of size n/mi .
b) for any k-tuple (B1 , B2 , . . . , Bk ), where Bi is a block of πi for
each i = 1, 2, . . . , k, we have

|B1 ∩ B2 ∩ · · · ∩ Bk | = 1.

◮ · We denote by N (k) the species of k-rectangles.


Cartesian Product
Species Rectangles

A Picture of a 3-Rectangle

A 3-rectangle on [24], labeled on
1111111111
0000000000
A1 A2 A3 A4 triangles.
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
◮ All triangles are labeled with
1111111111
0000000000 {1, 2, 3, . . . 24}.
1111111111
0000000000
B1 C2

1111111111
0000000000 ◮ π1 has 4 blocks, each of size 6
B2
1111111111
0000000000C
1111111111
0000000000
B3
1 ◮ π2 has 3 blocks, each of size 8
◮ π3 has 2 blocks, each of size 12
π1 = {A1 , A2 , A3 , A4 }
π2 = {B1 , B2 , B3 }
π3 = {C1 , C2 }


Cartesian Product
Species Rectangles

Definition

The following definition was given by Maia and Mendez (2006).
Definition
Let F1 and F2 be species of structures with F1 [∅] = F2 [∅] = ∅.
The arithmetic product of F1 and F2 , denoted F1 ⊡ F2 , is defined as

(F1 ⊡ F2 )[U] = F1 [π1 ] × F2 [π2 ],
(π1 π2 )∈N [U]

where the sum represents the disjoint union and U is a finite set.


Cartesian Product
Species Rectangles

Arithmetic Product

In other words, an F1 ⊡ F2 -structure on a ﬁnite set U is a tuple of
the form ((π1 , f1 ), (π2 , f2 )), where
◮ (π1 , π2 ) is a rectangle on the ﬁnite set U
◮ f1 is an F1 -structure on the blocks of π1
◮ f2 is an F2 -structure on the blocks of π2


Cartesian Product
Species Rectangles

A Picture: F1 ⊡ F2

X
X

X
F1 X
F2
X
X
X

=
X
X
X X
F1 X
X
X
F2
X
X


Cartesian Product
Species Rectangles

Properties of Arithmetic Product

The arithmetic product is

◮ commutative: F1 ⊡ F2 = F2 ⊡ F1
◮ associative: F1 ⊡ (F2 ⊡ F3 ) = (F1 ⊡ F2 ) ⊡ F3
◮ distributive: F1 ⊡ (F2 + F3 ) = F1 ⊡ F2 + F1 ⊡ F3
◮ with a unit X : F1 ⊡ X = X ⊡ F1 = F1


Cartesian Product
Species Rectangles

An Illustration of F ⊡ X = F

X
X
F X X =
F X

X X

◮ · An illustration of the equality

F ⊡ X = F.


Cartesian Product
Species Rectangles

Arithmetic Product of {F1 , F2 , . . . , Fk }

Definition
The arithmetic product of species F1 , F2 , . . . , Fk with Fi (∅) = ∅ for
all i is defined by setting
k
⊡ Fi = F1 ⊡ F2 ⊡ · · · ⊡ Fk ,
i =1

which sends each finite set U to the set
k
⊡ Fi [U] = F1 [π1 ] × F2 [π2 ] × · · · × Fk [πk ],
i =1
(π1 ,π2 ,...,πk )∈N (k) [U]

where the sum represents the disjoint union.
◮ · We denote by F ⊡k the arithmetic product of k copies of F .

Cartesian Product
Species

Outline
Species
Species of Graphs
Rectangles

Cartesian Product
Species

The Arithmetic Product E2 ⊡ X E2

◮ · The species E2 is the species associated to the graph
.
◮ · The species X E2 is the species associated to the graph
.
◮ · The species E2 ⊡ X E2 is the species associated to the
Cartesian product of and :

=


Cartesian Product
Species

This is Because..

◮ · Every automorphism of the product graph is generated by
automorphisms of the original graphs:


Cartesian Product
Species

Automorphism Group of Two Relatively Prime Graphs

◮ · Sabidussi proved that if G1 and G2 are relatively prime to each
other, then

aut(G1 ⊡ G2 ) ∼ aut(G1 ) × aut(G2 ).
=

◮ · That is,
the automorphism group of the product of two
relatively prime graphs is the product of the
automorphism groups of the graphs.


Cartesian Product
Species

The Arithmetic Product E2 ⊡ E2

◮ · The species E2 is the species associated to the graph
.
◮ · The species E2 ⊡ E2 is the species associated to the Cartesian
product of and :

=


Cartesian Product
Species

But..

◮ · Instead we want the species associated to the graph

in which the horizontal and vertical edges are not distinguished.


Cartesian Product
Species

Automorphism Group of P k

◮ · Let P be a prime graph. Let k be an integer.
◮ · The automorphism group of P k is, in fact, the so-called
exponentiation (Palmer, Robinson) of the symmetric group of
order k and the automorphism group of P.
◮ · In particular,
k
aut(P k ) = aut(P) .


Cartesian Product
Species

The species F ⊡k

◮ · Let F be a species of structures with F [∅] = ∅. Let k be a
positive integer.

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 ) ∈ N (k) [U] is a k-rectangle on U
◮ fi ∈ F [πi ] is an F -structure on the blocks of πi , for each
i ∈ [k]


Cartesian Product
Species

The Set E2⊡2[4]

Example
1 2 1 4 1 4

3 4 3 2 2 3

1 3 1 3 1 2

2 4 4 2 4 3

⊡2
There are 6 elements in the set E2 [4].


Cartesian Product
Species

A Group Action of Sk

◮ · The symmetric group Sk acts on the set F ⊡k [U] by permuting
the subscripts of πi and fi , i.e.,

α((π1 , f1 ), (π2 , f2 ), . . . , (πk , fk )) = ((πα1 , fα1 ), (πα2 , fα2 ), . . . , (παk , fαk )),

where
◮ α ∈ Sk is a permutation on [k]
◮ (πα1 , πα2 , . . . , παk ) ∈ N (k) [U]

◮ fαi ∈ F [παi ] for i ∈ [k].


Cartesian Product
Species

The Action of S2 on the Set E2⊡2 [4]

◮ · The
Example
transposition on the
1 2 1 4 1 4
subscripts of πi
3 4 3 2 2 3
switches rows and
columns.
◮ · This action
1 3 1 3 1 2 results in 3 orbits.

2 4 4 2 4 3


Cartesian Product
Species


Definition
We define the exponential composition of F of order k, denoted
Ek F , by setting for each finite set U, (Ek F )[U] be the set of
orbits of the action of Sk on F ⊡k [U].
◮ · We set E0 F = X .
Definition
We define exponential composition of F , denoted E F , to be the
sum of Ek F on all nonnegative integers k, i.e.,

E F = Ek F .
k≥0


Cartesian Product
Species

Connected Graphs and Prime Graphs

The unique factorization of connected graphs into products of
powers of prime graphs leads to the following theorem.

Theorem
The species C of connected graphs and Π of prime graphs satisfy

C =E Π .


Cartesian Product
Species

Enumeration Theorem

◮ · Palmer and Robinson (1973) proved a theorem which is useful
for computing the cycle index of the species of prime graphs.

Let A and B be permutation groups acting on [m]
and [n], respectively. Then the cycle index polynomial of
the exponentiation of A and B is the image of the cycle
index polynomial of B under the operator obtained by
substituting a certain kind of operators indexed by r for
the variables pr in the cycle index polynomial of A.


Cartesian Product
Species

The End

Thank you.


Arithmetic Product of Species

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