MATH 580, ALGEBRAIC GRAPH THEORY
Lecture 15 - Autormorphisms of Graphs
Reference: 1.3 - 1.6
Definition 1. Let X and Y be graphs. A mapping f : V (X) V (Y ) is an isomorphism if f is a
bijection and (x, y) is an edge in X if and only if (f (x), f (y) is an

Set partitions of [n]
Definition
A set partition of [n] is a collection = cfw_B1 , . . . , Bk of subsets of
[n] such that
Bi 6= ,
Bi Bj = ,
and
B1 Bk = [n].
The Bi are the blocks of .
Example
The set partitions of [3] are
1/2/3
12/3
Martha Yip
13/2
1/23

MATH 580, GRAPH THEORY TERMINOLOGY
1. Introduction
Definition 1. A graph X consists of a vertex set V (X) and an edge set E(X) of unordered pairs of vertices.
We will often use the notation xy to denote the edge cfw_x, y.
Definition 2. A graph is simple i

Sieve Method I: The Principle of Inclusion-Exclusion
The Principle of Inclusion-Exclusion
Let S be a set with n elements. Let V be the 2n -dimensional
vector space of all functions f : 2S k. Let : V V be the
linear transformation defined by
X
(f )(T ) =
f

Sieve Method I: The Principle of Inclusion-Exclusion
The basic idea for counting the objects in a set S is to first
over-count by considering a larger set, then subtracting the
unwanted objects.
Baby Example
Let A, B S. The number of elements of S not in

3.7 The Mobius function
Definition
The Mobius function of P is = 1 .
The values of can be calculated recursively:
for all x P,
(x, x) = 1
(x, y ) =
X
(x, z),
for all x y .
xz<y
In a locally finite poset P, if [a, b]
= [c, d], then (a, b) = (c, d).
Defin

Sieve Method II: Sign reversing involutions
The idea.
Let S be a set of objects with weight function w : S Z.
Suppose S = X
G B, and G is the set of objects we want to count;
(ie. we want
w (s). Then
sG
X
w (s) =
sS
If it happens that
X
X
w (s) +
sG
X
w (

What is enumerative combinatorics?
What does it mean to count something?
Martha Yip
Math 580, Lecture 1 Notes, based on Stanley 1.1 - 1.2
What is enumerative combinatorics?
What does it mean to count something?
There is no definitive answer.
- R. P. Stanl

3.1 The basics
Definition
A poset is a pair (P, ) where P is a set and is a relation on P
satisfying:
I
reflexivity: x x for all x P
I
transitivity: if x y and y z, then x z, and
I
antisymmetry: if x y and y x, then x = y .
Definition
For x, y P, say y co

Partitions of n
Definition
A partition of n is a sequence = (1 , . . . , k ) such that
X
i = n,
and
1 k > 0.
If is a partition of n, write ` n. The i are the parts of . If
has k nonzero parts, write `() = k.
Example
The partitions of n = 4 are
(4)
(3, 1)

3.1 Rank generating functions
Definition
Let P be a ranked poset with ranking function . If there are pk
elements of P with rank k, then the rank generating function of P
is
X
F (P; x) =
pk x k .
k0
Example
2.
3.
Pn
n k
k=0 k x , for the Boolean lattice,

MATH 580, POLYAS
THEOREM
Consider the following example before we proceed to the proof of the central theorem. We follow the same
notation as in the lecture.
Example 1. Consider the element 3 = (12)(34) D4 . The fixed points of the action of 3 on the 16

3.1 Rank generating functions
Definition
Let P be a ranked poset with ranking function . If there are pk
elements of P with rank k, then the rank generating function of P
is
X
F (P; x) =
pk x k .
k0
Example
2.
3.
Pn
n k
k=0 k x , for the Boolean lattice,

MATH 580 Test 1. Due Thursday October 25.
Rules:
1.
2.
3.
4.
5.
6.
If any question is unclear, email me.
You may look at Stanleys Enumerative Combinatorics.
You may look at any lecture slides and lecture notes.
You may refer to any old homework problems f

MATH 580 Problem Set 3. Due Tuesday October 16.
1. Show that a (b c) (a b) (a c), for any elements a, b, c in a lattice L.
2. Let L be a lattice. Show that the set of all sublattices of L, ordered by inclusion, is a complete
lattice.
3. Let f : P Q be an

MATH 580 Problem Set 1. Due Tuesday September 18.
1. Warm up with the following numerical problems.
(a) How many subsets of the set [10] = cfw_1, . . . , 10 contain at least one odd integer?
(b) In how many ways can seven people be seated in a circle if t

MATH 580 Problem Set 2. Due Tuesday October 2.
1. Compute the exponential generating series
X
dn
n0
xn
, where dn is the number of derangements
n!
of n.
k
nk
X
X
k
j nk
2. For 0 k n, prove that
dni =
(1)
(n j)!.
i
j
i=0
j=0
3. Give a combinatorial proof

3.4 Poset of order ideals
Definition
Given a poset P, let J(P) be the poset of order ideals of P,
ordered by inclusion.
Proposition
If P is a poset with n elements, then J(P) is ranked with rank
function
(I ) = |I |
for I J(P).
Proposition
The poset J(P)

MATH 580 Test 2. Due Monday December 17 5pm in my mailbox.
(Id greatly appreciate it if you can hand it in sooner.)
Rules:
1.
2.
3.
4.
5.
6.
If any question is unclear, email me.
You may look at Godsil and Royles Algebraic Graph Theory.
You may look at an

MATH 580 Problem Set 6. Due Tuesday November 27.
1. Let X be a graph. If S V (X), and (X\S) < (X), show that any retract of X contains at
least one vertex of S.
2. If X is arc-transitive, show that its core is arc-transitive.
3. Let X, Y be graphs. Show t

3.9 Rotas Closure Theorem
Definition
A closure operation on a poset is a map x 7 x such that
1. x x,
2. x = x,
3. if x y , then x y .
Let P = cfw_x P | x = x be the subposet of closed elements.
Theorem (Rota)
Let P be a locally finite poset with a closure

Counting with objects symmetry
Math 580 Lecture 14 notes
October 25 2012
Motivating problem
There are k colours of beads. How many different necklaces can
be made with n beads?
Let's begin with
n
= 4 and k = 2.
Label the beads:
x
x
1
2
4
3
x
x
1
Moti

MATH 580 Problem Set 5. Due Thursday November 15.
1. A graph X is self-complementary if X = X. Show that if there is a self-complementary graph on
n vertices, then n 0, 1 mod 4. If X is vertex regular, show that n 1 mod 4.
2. Let X be a (simple) graph on

MATH 580 Problem Set 4. Due Thursday November 1.
1. Calculate the cycle index polynomial ZCn (z1 , . . . , zn ) for the cylic group
Cn = h | n = 1i.
2. Find the generating series for the number of ways to paint the edges of the cube with red or
blue edges