Thesis Defense of Ji Li

Counting Point-Determining Graphs
and Prime Graphs
Using Joyal’s Theory of Species
Dissertation Defense
Thesis Advisor: Professor Ira Gessel

Ji Li

Department of Mathematics
Brandeis University
415 South Street, Waltham, MA

May 10th, 2007

Theory of Species Point-Determining Graphs Prime Graphs

Outline
1 Theory of Species
Deﬁnition of Species
Operations of Species
2 Point-Determining Graphs
Point-Determining Graphs
Bi-Point-Determining Graphs
Point-Determining 2-colored Graphs
3 Prime Graphs
Cartesian Product of Graphs
Molecular Species and P´lya’s Cycle Index Polynomial
o
Arithmetic Product of Species
Exponentiation Group
Exponential Composition of Species

J. L. Counting Point-Determining Graphs and Prime Gra


Definition of Species
Let B be the category of finite sets with bijections. A species (of
structures) is a functor
F :B→B
that generates for each finite set U a finite set F [U ], 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.

Unlabeled F -Structures
The symmetric group Sn acts on the set F [n] = F [{1, 2, . . . , n}] by
transport of structures. The Sn -orbits under this action are called
unlabeled F -structures of order n.



Species of Graphs
We denote by G the species of (simple) graphs. Then G [U ] is the set
of graphs with vertex set U

Example

1 3 5
U = {1, 2, 3, 4, 5}

2 4

σ

a c e

V = {a, b, c, d, e}
b d



Associated Series of Species
Each species F is associated with an exponential generating series
xn
F (x) = |F [n]| ,
n!
n≥0

a type generating series

F (x) = fn xn ,
n≥0

where fn is the number of unlabeled F -structures of order n, and a
cycle index of the species F , denoted ZF , satisfying

F (x) = ZF (x, 0, 0, . . . ), F (x) = ZF (x, x2 , x3 , . . . ).



Sum of Species
An F1 + F2 -structure on a ﬁnite set U is either an F1 -structure on U
or an F2 -structure on U .

= or

F1 F2
F1 + F2

Product of Species
An F1 F2 -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 , fi is an
Fi -structure on Ui for each i.

=
F1 · F2 F1 F2



Composition of Species
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 the blocks of π
• γ is a set of F2 -structures on each block of π.

F2
F2
= F2 = F2

F1 ◦ F2 F1 F2 F1 F2



Quotient Species
We say that a group A acts naturally on a species F , if for all 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:
ρU
A × F [U ] − − → F [U ]
−−
 
idA ×F [σ]
 F [σ]
ρV
A × F [V ] − − → F [V ]
−−
The quotient species of F by A, denoted F/A, is such that for any
ﬁnite set U ,
(F/A)[U ] = F [U ]/A.
In other words, the set of F/A-structures on U is the set of A-orbits
of F -structures on U .



Composition with Ek as a Quotient Species
Let k be any positive integer. Let Ek be the species of k-element sets.
Let
F · F · ····F
Fk = .
k copies
We observe that

F F F
F F F

Sk -orbits
Ek

F k /Sk = Ek ◦ F.



Neighborhood and Augmented Neighborhood
In a graph G, the neighborhood of a vertex v is the set of vertices
adjacent to v, the augmented neighborhood of a vertex is the union of
the vertex itself and its neighborhood.

Example

v

w1 w2 w3 w4

In the above ﬁgure, the neighborhood of the vertex v is the set
{w1 , w2 , w3 , w4 }, while the augmented neighborhood of v is the set
{v, w1 , w2 , w3 , w4 }.



Point-Determining Graphs
and Co-Point-Determining Graphs
• A graph is called point-determining if no two vertices of this
graph have the same neighborhoods.
• A graph is called co-point-determining if no two vertices of this
graph have the same augmented neighborhoods.

Example

The graph on the left is co-point-determining, and the graph on the
right is point-determining. These two graphs are complements of each
other.



A Natural Transformation
Let P be the species of point-determining graphs, and let Q be the
species of co-point-determining graphs. There is a natural
transformation
α:P →Q
that sends each point-determining graph to its complement, which is
a co-point-determining graph on the same vertex set, such that the
following diagram commutes for any bijection σ : U → V :
P[σ]
P[U ] − − → P[V ]
−−
 
α
 α

Q[σ]
Q[U ] − − → Q[V ]
−−
We call the species P isomorphic to the species Q, written as

P = Q.



Transform a Graph into a Point-Determining Graph

3 3

9 2 9 2

1 5 1 5

8 6 8 6

4 7 4 7

The transformation from a graph G with vertex set
[11] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} to a point-determining graph P
with vertex set {{1, 9, 3}, {8}, {4, 7}, {6}, {2, 5}}.



Generating Series of Point-Determining Graphs
Let E+ be the species of non-empty sets. We get a species identity

G = P ◦ E+ ,

which enables us to enumerate point-determining graphs. For
example, we can write down the beginning terms of the exponential
generating series and the type generating series of P (previously done
by Read):

x x2 x3 x4 x5 x6 x7
P(x) = 1+ + +4 +32 +588 +21476 +1551368 +· · ·
1! 2! 3! 4! 5! 6! 7!

P(x) = 1 + x + x2 + 2 x3 + 5 x4 + 16 x5 + 78 x6 + 588 x7 + · · ·



Bi-Point-Determining Graphs
We denote by R the species of bi-point-determining graphs, which are
graphs that are both point-determining and co-point-determining.

Example

Unlabeled bi-point-determining graphs with no more than 5 vertices.



Alternating Phylogenetic Trees
A phylogenetic tree is a rooted tree with labeled leaves and unlabeled
internal vertices in which no vertex has exactly one child.
An alternating phylogenetic tree is either a single vertex, or a
phylogenetic tree with more than one labeled vertex whose internal
vertices are colored black or white, where no two adjacent vertices are
colored the same way.

Example
5
4
8 An alternating
6
9 phylogenetic tree on 9
3
vertices, where the root
1
7
is colored black.
2



Transform a Graph into a Bi-Point-Determining Graph
6 3
6 3
8
8 5

5 1
2

4 7
1
2

4 7 6 8 2 5 1 4

On each step, we group vertices with the same neighborhoods
or vertices with the same augmented neighborhods.
6 3

Whenever vertices with the same neighborhods are grouped, 8

we connect the corresponding vertices/alternating phylogenetic 5

trees with a black node. 1
2

Whenever vertices with the same augmented neighborhoods 4 7

are grouped, we connected the corresponding vertices/
alternating phylogenetic trees with a white node. 7

6 8 2 5 1 4
Vertices left untouched are not colored.



A Species Identity for Bi-Point-Determining Graphs
The species of graphs is the composition of the species of
bi-point-determining graphs and the species of alternating
phylogenetic trees

T
= T
G
R T

G =R ◦T



Generating Series of Bi-Point-Determining Graphs
Through calculation, we write functional equations for the
exponential generating series and the type generating series of R:

x x4 x5 x6 x7 x8
R(x) = +12 +312 +13824 +1147488 +178672128 +· · ·
1! 4! 5! 6! 7! 8!

R(x) = x + x4 + 6x5 + 36x6 + 324x7 + 5280x8 + · · ·



Multisort Species
Let Bk be the category of ﬁnite k-sets with bijective multifunctions.
A species of k sorts is a functor

F : Bk → B.

2-Colored Graphs
A 2-colored graph is a graph in which all vertices are colored either
white or black, and no two adjacent vertices are assigned the same
color.
We denote by G (X, Y ) the 2-sort species of 2-colored graphs, where
vertices colored white are of sort X, and vertices colored black are of
sort Y .



Point-Determining 2-colored Graphs
• 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 graph is semi-point-determining, but it is
not point-determining.
We denote by
• P(X, Y ) — the 2-sort species of point-determining 2-colored
graphs
• P s (X, Y ) — the 2-sort species of semi-point-determining
2-colored graph
• P c (X, Y ) — the 2-sort species of connected point-determining
2-colored graph



Functional Equations: Part I
The idea is similar to the formula for the species of point-determining
graphs P:
G = P ◦ E+ .

We transform a 2-colored graph into a semi-point-determining
2-colored graph by grouping vertices with the same neighborhoods.
Note that if two vertices have the same neighborhoods, then they
must be colored in the same way.

G (X, Y ) = P s (E+ (X), E+ (Y )).



Functional Equations: Part II
The observation that a semi-point-determining graph consists of
• one or none isolated vertex colored white
• one or none isolated vertex colored black
• a set (possibly empty) of connected point-determining 2-colored
graphs with at least two vertices
leads to the functional equation:

P s (X, Y ) = (1 + X)(1 + Y ) E (P≥2 (X, Y ))
c

c c
P≥2 (X, Y ) P≥2 (X, Y )

1+X
P s (X, Y ) E
1+Y
c c
P≥2 (X, Y ) P≥2 (X, Y )



Functional Equations: Part III
Similarly, we observe that a point-determining 2-colored graph
consists of
• one or none isolated vertex, colored white or black
• a set of connected point-determining 2-colored graphs with at
least two vertices
Therefore,
c
P(X, Y ) = (1 + X + Y ) E (P≥2 (X, Y ))
c c
P≥2 (X, Y ) P≥2 (X, Y )

P(X, Y ) E
1+X +Y

c c
P≥2 (X, Y ) P≥2 (X, Y )



Generating Series of Point-Determining 2-colored Graphs
These functional equations allow us to calculate the generating series
of the species P s (X, Y ), P c (X, Y ), and P(X, Y ).

For example,

P(x, y) = 1 + x + y+
xy + x2 y + xy 2 + 2x2 y 2
+ 3x3 y 2 + 3x2 y 3 + · · · . Unlabeled point-determining
2-colored graphs with no more
than 5 vertices.



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 )},

and in which the vertex (u1 , v1 ) is adjacent to the vertex (u2 , v2 ) if
either u1 = u2 and v1 is adjacent to v2 or v1 = v2 and u1 is adjacent
to u2 .
Example
1
1,1’
2 4
2,1’ 4,1’

3
3,1’ 1,2’

2,2’ 4,2’
1,3’

1’ 2,3’ 4,3’
3,2’

3’ 2’ 3,3’



Properties of the Cartesian Product
The Cartesian product is commutative and associative. We write
n
Gn = ⊡ G.
i=1

Prime Graphs
A graph G is said to be prime with respect to Cartesian
multiplication if G is a non-trivial connected graph such that
G ∼ H1 ⊙ H2 implies that either H1 or H2 is a singleton vertex.
=

Relatively Prime
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 that J is a singleton vertex.



Decomposition of a Connected Graph
Any non-trivial connected graph can be decomposed into prime
factors. Sabidussi proved that such a prime factorization is unique up
to isomorphism.
Example

=

A connected graph with 24 vertices is decomposed into prime graphs
with 2 vertices 3 vertices, and 4 vertices, respectively.


Molecular Species
• A molecular species is a species that is indecomposable under
addition.
• If M is molecular, then M = Mn for some n, i.e., M [U ] is
nonempty if and only if U is an n-element set.
• If M = Mn , then M = X n /A for some subgroup A of Sn .
• The X n /A-structures on a ﬁnite set U , where |U | = n, is the set
of A-orbits of the action A on the set of linear orders on U . In
other words, X n /A is the quotient species of X n by A.
• Each subgroup A of Sn gives rise to a molecular species X n /A.



Cycle Index of a Group
• Let A be a subgroup of Sn . The cycle index polynomial of A,
deﬁned by P´lya, is
o
n
1 c (σ)
Z(A) = Z(A; p1 , p2 , . . . , pn ) = pkk ,
|A|
σ∈A k=1

where for a permutation σ, ck (σ) is the number of k-cycles in σ.
• If a molecular species M = X n /A, then the cycle index of the
species M is the same as the cycle index polynomial of the group
A. That is,
Z(A) = ZX n /A .



Species Associated to a Graph
Each graph G is associated to a species OG , where the OG -structures
on a finite set U is defined to be the set of graphs isomorphic to G
with vertex set U .

a b b c c d d e e a

d c e d a e b a c b
e a b c d

G OG [{a, b, c, d, e}]

OG is Molecular
The automorphism group of G acts on the vertex set of G. If G is a
graph with n vertices, then aut(G) may be identified with a subgroup
of Sn , and
Xn
OG = .
aut(G)



Product Group
Let A be a subgroup of Sm , and let B be a subgroup of Sn . We
deﬁne the product group A × B to be the subgroup of Smn such that
a) the group operation is

(a1 , b1 ) · (a2 , b2 ) = (a1 a2 , b1 b2 )

b
b) an element (a, b) of
A × B acts on (i, j) for
some i ∈ [m] and j ∈ [n] by

(a, b)(i, j) = (a(i), b(j)) a



Arithmetic Product of Species
In the above setting, we start with two molecular species X m /A and
X n /B, and get a new molecular species X mn /(A × B), which is
deﬁned to be the arithmetic product of X m /A and X n /B:

B-orbits

Xm Xn X mn
⊡ := .
A B A×B
A-orbits

The arithmetic product of species was previously studied by Maia and
M´ndez.
e



Properties of the Arithmetic Product
The arithmetic product has the following properties (given by Maia
and M´ndez):
e

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 .



and Arithmetic Product of Species
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
Since G1 and G2 are relatively prime, a theorem of Sabidussi gives
that aut(G1 ⊙ G2 ) = aut(G1 ) × aut(G2 ). Therefore,

X mn X mn
OG1 ⊙G2 = =
aut(G1 ⊙ G2 ) aut(G1 ) × aut(G2 )
Xm Xn
= ⊡ = OG1 ⊡ OG2 .
aut(G1 ) aut(G2 )



Exponentiation Group
Let A be a subgroup of Sm , and let B be a subgroup of Sn . The
exponentiation group B A is a subgroup of Snm , whose group
elements are of the form (α, τ ) with α ∈ A and τ : [m] → B.
a) The composition of
two elements (α, τ ) and

α
(β, η) is given by α τ (1)

(α, τ )(β, η) = (αβ, (τ ◦β)η).

τ (5 )

τ (2 )
b) The element (α, τ )
acts on the set of
functions from [m] to

α
[n] by sending each τ(
α

4) 3)
τ(
f : [m] → [n] to g,
where for all i ∈ [m],
α
g(i) = τ (i)(f (α−1 i)).


I Operators
Let (α, τ ) be an element of the exponentiation group B A such that
• α is an m-cycle in the group A
• τ = (τ (1), τ (2), . . . , τ (m)) ∈ B m satisﬁes that the cycle type of
τ (m)τ (m − 1) · · · τ (2)τ (1) is λ
Palmer and Robinson deﬁned the operators Im on the power sum
symmetric functions by
Im (pλ ) = pγ ,
where γ is the cycle type of the element (α, τ ) of B A .
More explicitly, γ = (γ1 , γ2 , . . . ) is the partition of nm with
gcd(m,l)
1 j
cj (γ) = µ ici (λ) .
j l
l|j i | l/ gcd(m,l)



⊠ Operator
• The operation ⊠ on the symmetric functions is deﬁned by letting

pν := pλ ⊠ pµ ,

where
ck (ν) = gcd(i, j) ci (λ)cj (µ).
lcm(i,j)=k

• If a ∈ A has cycle type λ, and b ∈ B has cycle type µ, then
(a, b) ∈ A × B has cycle type ν.
• If λ = (λ1 , λ2 , . . . ) is a partition of n, then

Iλ (pµ ) = Iλ1 (pµ ) ⊠ Iλ2 (pµ ) ⊠ · · · .



Cycle Index of Exponentiation Group
Theorem
(Palmer and Robinson) The cycle index polynomial of B A is the
image of Z(B) under the operator obtained by substituting the
operator Ir for the variables pr in Z(A). That is,

Z(B A ) = Z(A) ∗ Z(B).



An Example of the Exponentiation Group
Let A = S2 and B = C3 .
The element (α, τ ) of B A , with α = (1, 2), τ (1) = id and
τ (2) = (1, 2, 3), acts on the set of functions from [2] to [3].

The cycle
type of (α, τ )
is (6, 3),
which means,

I2 (p3 ) = p3 p6 .
We can calculate the cycle index of the exponentiation group using
Palmer and Robinson’s theorem:
1 9
Z(B A ) = (p + 8p3 + 3p3 p3 + 6p3 p6 ).
18 1 3 2



Exponential Composition of Species
We deﬁne the molecular B-orbits A-
m or
species X n /B A to be the or
bit
s bit
s
A-
exponential composition of
X m /A and X n /B:

B - or
bits
B - or

bits
Xm Xn XN
:= .
A B BA

A-or

bits
A-or
bits
Or equivalently,
B- s
⊡m or bit
or
Xm Xn Xn bit
s B-
:= A.
A B B A-orbits



Exponential Composition of a General Species
• Recall that the species of k-element sets Ek = X k /Sk . We call
Ek F the exponential composition of F of order k.
• The cycle index of the exponential composition is

ZEk X n /A = Z(Sk ) ∗ Z(A).

• Setting E0 F = X, we set

E F := Ek F .
k≥0



Properties of Exponential Composition
The exponential composition of species satisﬁes the additive
properties:
k
Ek F1 + F2 = Ei F1 ⊡ Ek−i F2 ,
i=0

E F1 + F2 = E F1 ⊡ E F2 .



Prime Power P k
Let P be any prime graph, and k any nonnegative integer.
Sabidussi showed that the automorphism group of P k is

aut(P k ) = aut(P )Sk .

Therefore,
k k
Xk Xn Xn Xn
Ek OP = = = = OP k .
Sk aut(P ) aut(P )Sk aut(P k )

E OP = X + OP + OP 2 + · · · .



Species of Prime Graphs
Let G c be the species of connected graphs. Let P be the species of
prime graphs. We can write it in terms of the sum of all prime
graphs, i.e., P = P OP .
We then apply the additive property of the exponential composition:

E P =E OP = ⊡ E OP = ⊡(X + OP + OP 2 + · · · ).
P P
P

This means that we get all connected graphs, since each connected
graph has a unique prime factorization (Sabidussi)! Therefore,
Theorem
E P = G c.



Cycle Index of the Species of Prime Graphs
In order to get a formula for the cycle index of the exponential
composition, we generalize Palmer and Robinson’s theorem for the
cycle index polynomial of the exponentiation group, and the cycle
index of the species of prime graphs can be then calculated, say, using
Maple:

1 2 1 2 3 1
ZP = p + p2 + p + p1 p2 + p3
2 1 2 3 1 3
35 4 7 2 7 1
+ p + p2 p2 + p1 p3 + p2 + p4
24 1 4 1 3 8 2 4
91 5 19 3 4 2 3
+ p + p p2 + p3 p3 + 5p1 p2 + p1 p4 + p2 p3 + p5
15 1 3 1 3 1 2
3 5
+ ···



Thank you!


Thesis Defense of Ji Li

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