Download Algebraic graph theory

Algebraic graph theory
Gabriel Coutinho
University of Waterloo
November 6th, 2013
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices
Gabriel Coutinho
Algebraic graph theory
2 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices
We can associate many matrices to a graph X .
Gabriel Coutinho
Algebraic graph theory
2 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices
We can associate many matrices to a graph X .
Gabriel Coutinho
Algebraic graph theory
2 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices
We can associate many matrices to a graph X .
Adjacency:



A(X ) = 


Gabriel Coutinho
0
1
0
1
0
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0
1
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1
Algebraic graph theory
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2 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices
We can associate many matrices to a graph X .
Adjacency:



A(X ) = 





Incidence: 


1
1
0
0
0
1
0
0
1
0
0
1
1
0
0
0
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1
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0












Gabriel Coutinho
Algebraic graph theory
2 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices
We can associate many matrices to a graph X .
Adjacency:



A(X ) = 





Incidence: 


1
1
0
0
0
1
0
0
1
0
0
1
1
0
0
0
1
0
0
1
0
0
1
0
1
0
0
1
1
0






Gabriel Coutinho
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1
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1
0
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0
1
0
1
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1
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1
0
1
0
0
0
1
1
0
0






Tutte:

0
x12
0
x14 0
 −x12
0
x23
0 x25

 0
−x
0
x
23
34 x35

 −x14
0
−x34 0
0
0
−x25 −x35 0
0
Algebraic graph theory






2 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - incidence matrix
I
The incidence matrix D of directed graphs is totally
unimodular.
Gabriel Coutinho
Algebraic graph theory
3 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - incidence matrix
I
The incidence matrix D of directed graphs is totally
unimodular.
I
Implies that LPs with coefficient matrix D are integral.
Gabriel Coutinho
Algebraic graph theory
3 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - incidence matrix
I
The incidence matrix D of directed graphs is totally
unimodular.
I
Implies that LPs with coefficient matrix D are integral.
I
Can be used to prove max-flow-min-cut theorem and other
results.
Gabriel Coutinho
Algebraic graph theory
3 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - incidence matrix
I
The incidence matrix D of directed graphs is totally
unimodular.
I
Implies that LPs with coefficient matrix D are integral.
I
Can be used to prove max-flow-min-cut theorem and other
results.
I
The matrix Q = DD T is called Laplacian of the underlying
undirected graph X .
Gabriel Coutinho
Algebraic graph theory
3 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - incidence matrix
I
The incidence matrix D of directed graphs is totally
unimodular.
I
Implies that LPs with coefficient matrix D are integral.
I
Can be used to prove max-flow-min-cut theorem and other
results.
I
The matrix Q = DD T is called Laplacian of the underlying
undirected graph X .
I
Using linear algebra, one can prove that the determinant of a
submatrix of Q counts the number of spanning trees of X .
Gabriel Coutinho
Algebraic graph theory
3 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices



Adjacency: 





Incidence: 


1
1
0
0
0
1
0
0
1
0
0
1
1
0
0
0
1
0
0
1
0
0
1
0
1
0
0
1
1
0






Gabriel Coutinho
0
1
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0
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0
1
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1
0
0




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Tutte:

0
x12
0
x14 0
 −x12
0
x23
0 x25

 0
−x
0
x
23
34 x35

 −x14
0
−x34 0
0
0
−x25 −x35 0
0
Algebraic graph theory






4 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Tutte matrix
I
It’s determinant is a formal expression on those variables.
Gabriel Coutinho
Algebraic graph theory
5 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Tutte matrix
I
It’s determinant is a formal expression on those variables.
I
The graph has a perfect matching if and only if this
determinant is not identically zero.
Gabriel Coutinho
Algebraic graph theory
5 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Tutte matrix
I
It’s determinant is a formal expression on those variables.
I
The graph has a perfect matching if and only if this
determinant is not identically zero.
I
This was used by Tutte to prove his famous theorem about
matchings.
Gabriel Coutinho
Algebraic graph theory
5 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Tutte matrix
I
It’s determinant is a formal expression on those variables.
I
The graph has a perfect matching if and only if this
determinant is not identically zero.
I
This was used by Tutte to prove his famous theorem about
matchings.
I
Can be used to provide state of the art algorithms to find
matchings.
Gabriel Coutinho
Algebraic graph theory
5 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Matrices



Adjacency: 





Incidence: 


1
1
0
0
0
1
0
0
1
0
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1
1
0
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0
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Gabriel Coutinho
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1
1
0
0






Tutte:

0
x12
0
x14 0
 −x12
0
x23
0 x25

 0
−x
0
x
23
34 x35

 −x14
0
−x34 0
0
0
−x25 −x35 0
0
Algebraic graph theory






6 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Adjacency matrix A
Entries of Ak count the number of walks of length k.
Gabriel Coutinho
Algebraic graph theory
7 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Adjacency matrix A
Entries of Ak count the number of walks of

0
 1

A=
 0
 1
0
Gabriel Coutinho
length
1 0
0 1
1 0
0 1
1 1
k.
1
0
1
0
0
Algebraic graph theory
0
1
1
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0
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7 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Adjacency matrix A
Entries of Ak count the number of walks of length

0 1 0
 1 0 1

A=
 0 1 0
 1 0 1
0 1 1



2 0 2 0 1
0 5
 0 3 1 2 1 
 5 2




A2 = 
A3 = 
 2 1 3 0 1 
 1 6
 0 2 0 2 1 
 4 1
1 1 1 1 2
2 4
Gabriel Coutinho
k.
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Algebraic graph theory

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






7 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Properties - Adjacency matrix A
Entries of Ak count the number of walks of length

0 1 0
 1 0 1

A=
 0 1 0
 1 0 1
0 1 1



2 0 2 0 1
0 5
 0 3 1 2 1 
 5 2




A2 = 
A3 = 
 2 1 3 0 1 
 1 6
 0 2 0 2 1 
 4 1
1 1 1 1 2
2 4
k.
1
0
1
0
0
0
1
1
0
0

1
6
2
5
4
4
1
5
0
2
2
4
4
2
2









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
Can model random walks. Second largest eigenvalue determines how fast
a random walk becomes really random. Applications in cryptography.
Gabriel Coutinho
Algebraic graph theory
7 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph
I
An automorphism of a graph X is a bijection
ϕ : V (X ) → V (X ) that preserves adjacency.
Gabriel Coutinho
Algebraic graph theory
8 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph
I
An automorphism of a graph X is a bijection
ϕ : V (X ) → V (X ) that preserves adjacency.
I
Automorphisms of a graph forms a group, the Aut(X ).
Gabriel Coutinho
Algebraic graph theory
8 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph
I
An automorphism of a graph X is a bijection
ϕ : V (X ) → V (X ) that preserves adjacency.
I
Automorphisms of a graph forms a group, the Aut(X ).
Gabriel Coutinho
Algebraic graph theory
8 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph
I
An automorphism of a graph X is a bijection
ϕ : V (X ) → V (X ) that preserves adjacency.
I
Automorphisms of a graph forms a group, the Aut(X ).
In this example, Aut = D3 × Z2
Gabriel Coutinho
Algebraic graph theory
8 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph 2
I
Almost all graphs have a trivial automorphism group... (Erdős
and Rényi)
Gabriel Coutinho
Algebraic graph theory
9 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph 2
I
Almost all graphs have a trivial automorphism group... (Erdős
and Rényi)
I
... but every finite group can be realized as the automorphism
group of a finite graph. (Frucht)
Gabriel Coutinho
Algebraic graph theory
9 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph 2
I
Almost all graphs have a trivial automorphism group... (Erdős
and Rényi)
I
... but every finite group can be realized as the automorphism
group of a finite graph. (Frucht)
I
Group theory machinery can be used to answer problems in
graph theory.
Gabriel Coutinho
Algebraic graph theory
9 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph 2
I
Almost all graphs have a trivial automorphism group... (Erdős
and Rényi)
I
... but every finite group can be realized as the automorphism
group of a finite graph. (Frucht)
I
Group theory machinery can be used to answer problems in
graph theory.
I
Into how many (distinct!) ways can you colour the cycle C4
with 10 colours?
Gabriel Coutinho
Algebraic graph theory
9 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph 2
I
Almost all graphs have a trivial automorphism group... (Erdős
and Rényi)
I
... but every finite group can be realized as the automorphism
group of a finite graph. (Frucht)
I
Group theory machinery can be used to answer problems in
graph theory.
I
Into how many (distinct!) ways can you colour the cycle C4
with 10 colours?
Gabriel Coutinho
Algebraic graph theory
9 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Symmetries of a graph 2
I
Almost all graphs have a trivial automorphism group... (Erdős
and Rényi)
I
... but every finite group can be realized as the automorphism
group of a finite graph. (Frucht)
I
Group theory machinery can be used to answer problems in
graph theory.
I
Into how many (distinct!) ways can you colour the cycle C4
with 10 colours?
I
Using something called Burnside Lemma, we can (easily) find
the answer: 1540.
Gabriel Coutinho
Algebraic graph theory
9 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs
Groups can also be used to construct graphs. Pick a group G and
a subset S (closed under inverses).
Gabriel Coutinho
Algebraic graph theory
10 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs
Groups can also be used to construct graphs. Pick a group G and
a subset S (closed under inverses).
Define a graph with vertex set G , and “edge set” S, denoted by
Cay(G , S).
Gabriel Coutinho
Algebraic graph theory
10 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs
Groups can also be used to construct graphs. Pick a group G and
a subset S (closed under inverses).
Define a graph with vertex set G , and “edge set” S, denoted by
Cay(G , S).
In this example, G is the Quaternion Group {±1, ±i, ±j, ±k} and
S = {i, −i, j, −j}.
Gabriel Coutinho
Algebraic graph theory
10 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs 2
I
Cayley Graphs are highly symmetric. In fact,
G 6 Aut(Cay(G , S)) for any S.
Gabriel Coutinho
Algebraic graph theory
11 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs 2
I
Cayley Graphs are highly symmetric. In fact,
G 6 Aut(Cay(G , S)) for any S.
I
Graph theory tools can be used to study groups.
Gabriel Coutinho
Algebraic graph theory
11 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs 2
I
Cayley Graphs are highly symmetric. In fact,
G 6 Aut(Cay(G , S)) for any S.
I
Graph theory tools can be used to study groups.
I
Two major open problems:
Gabriel Coutinho
Algebraic graph theory
11 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs 2
I
Cayley Graphs are highly symmetric. In fact,
G 6 Aut(Cay(G , S)) for any S.
I
Graph theory tools can be used to study groups.
I
Two major open problems:
I
Ramsey theory questions about Cayley graphs (famous
conjecture by Noga Alon).
Gabriel Coutinho
Algebraic graph theory
11 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Cayley Graphs 2
I
Cayley Graphs are highly symmetric. In fact,
G 6 Aut(Cay(G , S)) for any S.
I
Graph theory tools can be used to study groups.
I
Two major open problems:
I
Ramsey theory questions about Cayley graphs (famous
conjecture by Noga Alon).
I
Is every Cayley Graph (apart from K2 ) Hamiltonian?
I
More general conjecture by Lovász, but this has received more
attention. It is solved for some classes of groups.
Gabriel Coutinho
Algebraic graph theory
11 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Polynomials associated to a graph
There are many polynomials associated to a graph X with n
vertices.
Gabriel Coutinho
Algebraic graph theory
12 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Polynomials associated to a graph
There are many polynomials associated to a graph X with n
vertices.
I
Characteristic polynomial of A(X ): φX (t) = det(tI − A(X )).
Gabriel Coutinho
Algebraic graph theory
12 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Polynomials associated to a graph
There are many polynomials associated to a graph X with n
vertices.
I
Characteristic polynomial of A(X ): φX (t) = det(tI − A(X )).
I
Matching polynomial µX (t). Coefficient of x k (up to signing)
counts the number of matchings incident to n − k vertices.
Gabriel Coutinho
Algebraic graph theory
12 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Polynomials associated to a graph
There are many polynomials associated to a graph X with n
vertices.
I
Characteristic polynomial of A(X ): φX (t) = det(tI − A(X )).
I
Matching polynomial µX (t). Coefficient of x k (up to signing)
counts the number of matchings incident to n − k vertices.
I
Tutte polynomial. Two variables. Defined for multigraphs
recursively as:
TX = TX \e + TX /e
for any e ∈ E (X )
and TX (x, y ) = x i y j is X has i bridges, j loops, and no other edges.
Gabriel Coutinho
Algebraic graph theory
12 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Polynomials associated to a graph
There are many polynomials associated to a graph X with n
vertices.
I
Characteristic polynomial of A(X ): φX (t) = det(tI − A(X )).
I
Matching polynomial µX (t). Coefficient of x k (up to signing)
counts the number of matchings incident to n − k vertices.
I
Tutte polynomial. Two variables. Defined for multigraphs
recursively as:
TX = TX \e + TX /e
for any e ∈ E (X )
and TX (x, y ) = x i y j is X has i bridges, j loops, and no other edges.
As opposed to what happened with matrices, none of these polynomials
actually determine the graph.
Gabriel Coutinho
Algebraic graph theory
12 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Examples
φ(t) = t 5 − 4t 3
µ(t) = t 5 − 4t 3
T (x, y ) = x 4
Gabriel Coutinho
Algebraic graph theory
13 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Examples
φ(t) = t 5 − 4t 3
µ(t) = t 5 − 4t 3
T (x, y ) = x 4
Gabriel Coutinho
φ(t) = t 5 − 4t 3 + 2t
µ(t) = t 5 − 4t 3 + 2t
T (x, y ) = x 4
Algebraic graph theory
13 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Examples
φ(t) = t 5 − 4t 3
µ(t) = t 5 − 4t 3
T (x, y ) = x 4
φ(t) = t 5 − 4t 3 + 2t
µ(t) = t 5 − 4t 3 + 2t
T (x, y ) = x 4
φ(t) = t 5 − 4t 3
µ(t) = t 5 − 4t 3 + 2t
T (x, y ) = y + x + x 2 + x 3
Gabriel Coutinho
Algebraic graph theory
13 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Reconstruction
I
The deck of a graph X is the set of subgraphs of X defined as
{X \u : u ∈ V (X )}.
Gabriel Coutinho
Algebraic graph theory
14 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Reconstruction
I
The deck of a graph X is the set of subgraphs of X defined as
{X \u : u ∈ V (X )}.
I
Kelly and Ulam conjectured that graphs with more than 2
vertices are determined from their deck.
Gabriel Coutinho
Algebraic graph theory
14 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Reconstruction
I
The deck of a graph X is the set of subgraphs of X defined as
{X \u : u ∈ V (X )}.
I
Kelly and Ulam conjectured that graphs with more than 2
vertices are determined from their deck.
I
Open Problem Garden thinks that it is a very important
conjecture in graph theory.
Gabriel Coutinho
Algebraic graph theory
14 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Reconstruction
I
The deck of a graph X is the set of subgraphs of X defined as
{X \u : u ∈ V (X )}.
I
Kelly and Ulam conjectured that graphs with more than 2
vertices are determined from their deck.
I
Open Problem Garden thinks that it is a very important
conjecture in graph theory.
I
Both the characteristic and the Tutte polynomial of a graph
are reconstructible from its deck.
Gabriel Coutinho
Algebraic graph theory
14 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Reconstruction
I
The deck of a graph X is the set of subgraphs of X defined as
{X \u : u ∈ V (X )}.
I
Kelly and Ulam conjectured that graphs with more than 2
vertices are determined from their deck.
I
Open Problem Garden thinks that it is a very important
conjecture in graph theory.
I
Both the characteristic and the Tutte polynomial of a graph
are reconstructible from its deck.
I
So the conjectured needs to be proved “only” for graphs with
similar such polynomials.
Gabriel Coutinho
Algebraic graph theory
14 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Tutte polynomial
I
By imposing relations on the variables x and y , TX (x, y ) can
have a good number of different interpretations.
Gabriel Coutinho
Algebraic graph theory
15 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Tutte polynomial
I
By imposing relations on the variables x and y , TX (x, y ) can
have a good number of different interpretations.
I
Up to a sign, TX (k − 1, 0) counts the number of k-proper
colourings in the graph.
Gabriel Coutinho
Algebraic graph theory
15 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Tutte polynomial
I
By imposing relations on the variables x and y , TX (x, y ) can
have a good number of different interpretations.
I
Up to a sign, TX (k − 1, 0) counts the number of k-proper
colourings in the graph.
I
In particular, computing the Tutte polynomial is hard.
Gabriel Coutinho
Algebraic graph theory
15 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Tutte polynomial
I
By imposing relations on the variables x and y , TX (x, y ) can
have a good number of different interpretations.
I
Up to a sign, TX (k − 1, 0) counts the number of k-proper
colourings in the graph.
I
In particular, computing the Tutte polynomial is hard.
I
Along the hyperbole xy = 1, TX is (up to factors) the Jones
polynomial of the corresponding alternating knot.
Gabriel Coutinho
Algebraic graph theory
15 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Matrices
Groups
Polynomials
Tutte polynomial
I
By imposing relations on the variables x and y , TX (x, y ) can
have a good number of different interpretations.
I
Up to a sign, TX (k − 1, 0) counts the number of k-proper
colourings in the graph.
I
In particular, computing the Tutte polynomial is hard.
I
Along the hyperbole xy = 1, TX is (up to factors) the Jones
polynomial of the corresponding alternating knot.
I
Strongly related to certain models of statistical physics.
Gabriel Coutinho
Algebraic graph theory
15 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
More regular than just regular
Look to the following graph.
Gabriel Coutinho
Algebraic graph theory
16 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
More regular than just regular
Look to the following graph.
Even though it is a regular graph, and actually quite symmetric,
the following two pairs of adjacent vertices in green seem to
interact differently:
Gabriel Coutinho
Algebraic graph theory
16 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
More regular than just regular
Look to the following graph.
Even though it is a regular graph, and actually quite symmetric,
the following two pairs of adjacent vertices in green seem to
interact differently:
Gabriel Coutinho
Algebraic graph theory
16 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
More regular than just regular
Look to the following graph.
Even though it is a regular graph, and actually quite symmetric,
the following two pairs of adjacent vertices in green seem to
interact differently:
The second pair has a common a neighbour.
Gabriel Coutinho
Algebraic graph theory
16 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs
A graph X is called strongly regular with parameters (n, k, a, c) if:
Gabriel Coutinho
Algebraic graph theory
17 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs
A graph X is called strongly regular with parameters (n, k, a, c) if:
I
it has n vertices and is k-regular.
Gabriel Coutinho
Algebraic graph theory
17 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs
A graph X is called strongly regular with parameters (n, k, a, c) if:
I
it has n vertices and is k-regular.
I
any pair of adjacent vertices has a common neighbours.
Gabriel Coutinho
Algebraic graph theory
17 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs
A graph X is called strongly regular with parameters (n, k, a, c) if:
I
it has n vertices and is k-regular.
I
any pair of adjacent vertices has a common neighbours.
I
any pair of non-adjacent vertices has c common neighbours.
Gabriel Coutinho
Algebraic graph theory
17 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs
A graph X is called strongly regular with parameters (n, k, a, c) if:
I
it has n vertices and is k-regular.
I
any pair of adjacent vertices has a common neighbours.
I
any pair of non-adjacent vertices has c common neighbours.
(5,2,0,1)
They arise naturally in the study
of finite geometries, design theory,
equiangular lines and group theory.
(6,4,2,4)
Gabriel Coutinho
Algebraic graph theory
17 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
Gabriel Coutinho
k(k − a − 1) = (n − k − 1)c
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
k(k − a − 1) = (n − k − 1)c
Where is the algebra? Crash course for the 1st comps....
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
k(k − a − 1) = (n − k − 1)c
Where is the algebra? Crash course for the 1st comps....
I
Consider A = A(X ). Note that A(X ) = J − I − A.
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
k(k − a − 1) = (n − k − 1)c
Where is the algebra? Crash course for the 1st comps....
I
Consider A = A(X ). Note that A(X ) = J − I − A.
I
Recall that powers of A count walks between vertices.
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
k(k − a − 1) = (n − k − 1)c
Where is the algebra? Crash course for the 1st comps....
I
Consider A = A(X ). Note that A(X ) = J − I − A.
I
Recall that powers of A count walks between vertices.
I
What is (A2 )u,u ? Valency of the graph: k.
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
k(k − a − 1) = (n − k − 1)c
Where is the algebra? Crash course for the 1st comps....
I
Consider A = A(X ). Note that A(X ) = J − I − A.
I
Recall that powers of A count walks between vertices.
I
What is (A2 )u,u ? Valency of the graph: k.
I
If u ∼ v , then (A2 )u,v = a.
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
k(k − a − 1) = (n − k − 1)c
Where is the algebra? Crash course for the 1st comps....
I
Consider A = A(X ). Note that A(X ) = J − I − A.
I
Recall that powers of A count walks between vertices.
I
What is (A2 )u,u ? Valency of the graph: k.
I
If u ∼ v , then (A2 )u,v = a.
I
If u v , then (A2 )u,v = c.
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 2
Typical problem about strongly regular graphs:
I
Which parameters correspond to an actual graph?
Easy constraints:
n >k −1 ;
k >a;
k ≥c ;
k(k − a − 1) = (n − k − 1)c
Where is the algebra? Crash course for the 1st comps....
I
Consider A = A(X ). Note that A(X ) = J − I − A.
I
Recall that powers of A count walks between vertices.
I
What is (A2 )u,u ? Valency of the graph: k.
I
If u ∼ v , then (A2 )u,v = a.
I
If u v , then (A2 )u,v = c.
I
So A2 = kI + aA + c(J − I − A).
Gabriel Coutinho
Algebraic graph theory
18 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 3
I
A2 = kI + aA + c(J − I − A).
Gabriel Coutinho
Algebraic graph theory
19 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 3
I
A2 = kI + aA + c(J − I − A).
I
Can you tell who are the eigenvalues of A?
Gabriel Coutinho
Algebraic graph theory
19 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 3
I
A2 = kI + aA + c(J − I − A).
I
Can you tell who are the eigenvalues of A?
I
If a graph is k-regular, the all 1s vector 1 is an eigenvector
with eigenvalue k.
Gabriel Coutinho
Algebraic graph theory
19 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 3
I
A2 = kI + aA + c(J − I − A).
I
Can you tell who are the eigenvalues of A?
I
If a graph is k-regular, the all 1s vector 1 is an eigenvector
with eigenvalue k.
I
Every other eigenvector is orthogonal to 1 (A is symmetric
and should remember this from your 1st linear algebra course)
Gabriel Coutinho
Algebraic graph theory
19 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 3
I
A2 = kI + aA + c(J − I − A).
I
Can you tell who are the eigenvalues of A?
I
If a graph is k-regular, the all 1s vector 1 is an eigenvector
with eigenvalue k.
I
Every other eigenvector is orthogonal to 1 (A is symmetric
and should remember this from your 1st linear algebra course)
−
If →
x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
I
Gabriel Coutinho
Algebraic graph theory
19 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 3
I
A2 = kI + aA + c(J − I − A).
I
Can you tell who are the eigenvalues of A?
I
If a graph is k-regular, the all 1s vector 1 is an eigenvector
with eigenvalue k.
I
I
Every other eigenvector is orthogonal to 1 (A is symmetric
and should remember this from your 1st linear algebra course)
−
If →
x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
I
Hence
I
−
−
−
−
A2 →
x = kI →
x + aA→
x + c(J − I − A)→
x ⇒ λ2 = k + aλ − c − cλ
p
a − c ± (c − a)2 + 4(k − c)
So λ =
2
Gabriel Coutinho
Algebraic graph theory
19 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 4
I
So λ =
a−c ±
p
(c − a)2 + 4(k − c)
.
2
Gabriel Coutinho
Algebraic graph theory
20 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 4
I
I
p
(c − a)2 + 4(k − c)
.
2
So you have three eigenvalues: k, θ, and τ .
So λ =
a−c ±
Gabriel Coutinho
Algebraic graph theory
20 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 4
p
I
(c − a)2 + 4(k − c)
.
2
So you have three eigenvalues: k, θ, and τ .
I
The sum of their multiplicities is the order of A, so n.
I
So λ =
a−c ±
Gabriel Coutinho
Algebraic graph theory
20 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 4
p
I
(c − a)2 + 4(k − c)
.
2
So you have three eigenvalues: k, θ, and τ .
I
The sum of their multiplicities is the order of A, so n.
I
The sum of all the eigenvalues (counting with multiplicities) is
the trace of A, so 0.
I
So λ =
a−c ±
Gabriel Coutinho
Algebraic graph theory
20 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 4
p
I
(c − a)2 + 4(k − c)
.
2
So you have three eigenvalues: k, θ, and τ .
I
The sum of their multiplicities is the order of A, so n.
I
The sum of all the eigenvalues (counting with multiplicities) is
the trace of A, so 0.
I
Easy to see: mk = 1.
I
So λ =
a−c ±
Gabriel Coutinho
Algebraic graph theory
20 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 4
p
I
(c − a)2 + 4(k − c)
.
2
So you have three eigenvalues: k, θ, and τ .
I
The sum of their multiplicities is the order of A, so n.
I
The sum of all the eigenvalues (counting with multiplicities) is
the trace of A, so 0.
I
Easy to see: mk = 1.
I
So 1 + mθ + mτ = n and k + θmθ + τ mτ = 0.
I
So λ =
a−c ±
Gabriel Coutinho
Algebraic graph theory
20 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Strongly regular graphs 4
p
I
(c − a)2 + 4(k − c)
.
2
So you have three eigenvalues: k, θ, and τ .
I
The sum of their multiplicities is the order of A, so n.
I
The sum of all the eigenvalues (counting with multiplicities) is
the trace of A, so 0.
I
Easy to see: mk = 1.
I
So 1 + mθ + mτ = n and k + θmθ + τ mτ = 0.
I
You can solve the system. For example, if
(n, k, a, c) = (58, 30, 20, 10), you get:
I
So λ =
a−c ±
mθ =
Gabriel Coutinho
√ 3
19 − 7 5
2
Algebraic graph theory
20 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Triangle free - Some examples
There are more algebraic tools that can be used to rule out
parameter sets. But we can’t go that far:
Gabriel Coutinho
Algebraic graph theory
21 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Triangle free - Some examples
There are more algebraic tools that can be used to rule out
parameter sets. But we can’t go that far:
I No one knows whether or not a srg (65,32,15,16) exists.
Gabriel Coutinho
Algebraic graph theory
21 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Triangle free - Some examples
There are more algebraic tools that can be used to rule out
parameter sets. But we can’t go that far:
I No one knows whether or not a srg (65,32,15,16) exists.
So let’s look to simpler cases. Suppose a = 0.
Gabriel Coutinho
Algebraic graph theory
21 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Triangle free - Some examples
There are more algebraic tools that can be used to rule out
parameter sets. But we can’t go that far:
I No one knows whether or not a srg (65,32,15,16) exists.
So let’s look to simpler cases. Suppose a = 0.
I Means adjacent vertices share no neighbours in common.
Aka: no triangles.
Gabriel Coutinho
Algebraic graph theory
21 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Open cases
The list of feasible parameter sets up to 300 vertices is:
Gabriel Coutinho
Algebraic graph theory
22 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Open cases
The list of feasible parameter sets up to 300 vertices is:
(5,2,0,1) - Pentagon.
(10,3,0,1) - Petersen Graph.
(16,5,0,2) - Clebsch Graph.
(50,7,0,1) - Hoffman-Singleton Graph.
(56,10,0,2) - Gewirtz Graph.
(77,16,0,4) - M22 Graph.
(100,22,0,6) - Mesner-Higman-Sims Graph.
(162,21,0,3) - ??
(176,25,0,4) - ??
(210,33,0,6) - ??
(266,45,0,9) - ??
Gabriel Coutinho
Algebraic graph theory
22 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Starting
Triangle free
Open cases
The list of feasible parameter sets up to 300 vertices is:
(5,2,0,1) - Pentagon.
(10,3,0,1) - Petersen Graph.
(16,5,0,2) - Clebsch Graph.
(50,7,0,1) - Hoffman-Singleton Graph.
(56,10,0,2) - Gewirtz Graph.
(77,16,0,4) - M22 Graph.
(100,22,0,6) - Mesner-Higman-Sims Graph.
(162,21,0,3) - ??
(176,25,0,4) - ??
(210,33,0,6) - ??
(266,45,0,9) - ??
If you require c = 1, Hoffman-Singleton theorem says that other than those three, you
can only have:
(3250,57,0,1) - major problem!
Gabriel Coutinho
Algebraic graph theory
22 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Random walks
If X is a graph with adjacency matrix A, we can scale the columns of A
e in such a way that A
e represents a probability distribution of a
to A
particle moving in the graph.
Gabriel Coutinho
Algebraic graph theory
23 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Random walks
If X is a graph with adjacency matrix A, we can scale the columns of A
e in such a way that A
e represents a probability distribution of a
to A
particle moving in the graph.
0
 1/2

 0
 1/2
0

1/3
0
1/3
0
1/3
0
1/3
0
1/3
1/3
Gabriel Coutinho
1/2
0
1/2
0
0
0
1/2
1/2
0
0



0
1/3
 1 
 0 




  0  =  1/3 
 0 
 0 
0
1/3

Algebraic graph theory
23 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Random walks
If X is a graph with adjacency matrix A, we can scale the columns of A
e in such a way that A
e represents a probability distribution of a
to A
particle moving in the graph.
0
 1/2

 0
 1/2
0

1/3
0
1/3
0
1/3
0
1/3
0
1/3
1/3
1/2
0
1/2
0
0
0
1/2
1/2
0
0



0
1/3
 1 
 0 




  0  =  1/3 
 0 
 0 
0
1/3

Powers of the scaled matrix represent the probability distribution of a
longer walk...
 5/12

 1/6 1/6 1/6 1/6
0
5/18
0
1/6
e2 = 
A
0
5/12
0
1/6
4/9
1/6
5/18
1/9
1/6
4/9
0
1/9
5/12
0
5/12
1/6
1/6
1/6
1/6
1/3
Gabriel Coutinho

e 100 ≈ 
A
1/4
1/4
1/6
1/6
1/4
1/4
1/6
1/6
Algebraic graph theory
1/4
1/4
1/6
1/6
1/4
1/4
1/6
1/6
1/6
1/4
1/4
1/6
1/6


23 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Quantum walks
I
Suppose that instead of classical particles, vertices of the
graph represent interacting quantum particles.
Gabriel Coutinho
Algebraic graph theory
24 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Quantum walks
I
Suppose that instead of classical particles, vertices of the
graph represent interacting quantum particles.
I
Quantum analogs of classical processes can have funny
behaviours.
Gabriel Coutinho
Algebraic graph theory
24 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Quantum walks
I
Suppose that instead of classical particles, vertices of the
graph represent interacting quantum particles.
I
Quantum analogs of classical processes can have funny
behaviours.
I
For example, the probability distribution does not converge.
Gabriel Coutinho
Algebraic graph theory
24 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Quantum walks
I
Suppose that instead of classical particles, vertices of the
graph represent interacting quantum particles.
I
Quantum analogs of classical processes can have funny
behaviours.
I
For example, the probability distribution does not converge.
I
But at some particular time, it might get concentrated...
Gabriel Coutinho
Algebraic graph theory
24 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Quantum walks 2
I
We use a continuous-time.
Gabriel Coutinho
Algebraic graph theory
25 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Quantum walks 2
I
We use a continuous-time.
I
Quantum theory says that the transition matrix is:
exp(−ıtA) = I + −ıtA −
Gabriel Coutinho
t 2 A2 ıt 3 A3 t 4 A4
+
+
− .....
2
3!
4!
Algebraic graph theory
25 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Quantum walks 2
I
We use a continuous-time.
I
Quantum theory says that the transition matrix is:
exp(−ıtA) = I + −ıtA −
I
t 2 A2 ıt 3 A3 t 4 A4
+
+
− .....
2
3!
4!
Question: how do you compute that?
Gabriel Coutinho
Algebraic graph theory
25 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
Gabriel Coutinho
0
1
1
0
.
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Gabriel Coutinho
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
Gabriel Coutinho
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=0
Gabriel Coutinho
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=
Gabriel Coutinho
π
6
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=
Gabriel Coutinho
π
4
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=
Gabriel Coutinho
π
3
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=
Gabriel Coutinho
π
2
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=
Gabriel Coutinho
π
2
+
π
6
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=
Gabriel Coutinho
π
2
+
π
4
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=
Gabriel Coutinho
π
2
+
π
3
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Example
Suppose your graph is K2 . Then A(K2 ) =
0
1
1
0
.
A very nice thing about this matrix is that:
Ak = A if k is odd
and Ak = I if k is even.
Hence
exp(−ıtA) = cos(t)I − ı sin(t)A =
cos(t)
−ı sin(t)
−ı sin(t)
cos(t)
The 1st column represents the quantum state at time t of the first
vertex. A quantum measurement in that column will say that we are at
the first vertex with probability cos(t)2 and at the second vertex with
probability sin(t)2 . Therefore:
t=π
Gabriel Coutinho
Algebraic graph theory
26 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool
I
The adjacency matrix of a graph is symmetric.
Gabriel Coutinho
Algebraic graph theory
27 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool
I
The adjacency matrix of a graph is symmetric.
I
Symmetric matrices can be orthogonally diagonalizable.
Gabriel Coutinho
Algebraic graph theory
27 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool
I
The adjacency matrix of a graph is symmetric.
I
Symmetric matrices can be orthogonally diagonalizable.
I
Suppose the distinct eigenvalues of A are {θ0 , ..., θd }.
Gabriel Coutinho
Algebraic graph theory
27 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool
I
The adjacency matrix of a graph is symmetric.
I
Symmetric matrices can be orthogonally diagonalizable.
I
Suppose the distinct eigenvalues of A are {θ0 , ..., θd }.
I
Let Ek be a projector for the eigenspace of θk .
Gabriel Coutinho
Algebraic graph theory
27 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool
I
The adjacency matrix of a graph is symmetric.
I
Symmetric matrices can be orthogonally diagonalizable.
I
Suppose the distinct eigenvalues of A are {θ0 , ..., θd }.
I
Let Ek be a projector for the eigenspace of θk .
I
Then
A=
d
X
θr E r
r =0
with
Er2
= Er , Er Es = 0 and
Gabriel Coutinho
P
Er = I .
Algebraic graph theory
27 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool 2
A very good feature of the spectral decomposition is that if:
Gabriel Coutinho
Algebraic graph theory
28 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool 2
A very good feature of the spectral decomposition is that if:
A=
d
X
θr E r
r =0
Gabriel Coutinho
Algebraic graph theory
28 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool 2
A very good feature of the spectral decomposition is that if:
A=
d
X
θr E r
r =0
then
k
A =
d
X
θrk Er
r =0
Gabriel Coutinho
Algebraic graph theory
28 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
In general - main tool 2
A very good feature of the spectral decomposition is that if:
A=
d
X
θr E r
r =0
then
k
A =
d
X
θrk Er
r =0
And then it is not difficult to convince yourself that:
exp(−ıtA) =
d
X
e −ıtθr Er
r =0
Gabriel Coutinho
Algebraic graph theory
28 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Perfect state transfer
A graph X admits perfect state transfer between vertices u and v
if there is a time t such that:
exp(−ıtA(X ))eu = λev .
Gabriel Coutinho
Algebraic graph theory
29 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Perfect state transfer
A graph X admits perfect state transfer between vertices u and v
if there is a time t such that:
exp(−ıtA(X ))eu = λev .
As we saw, K2 admits perfect state transfer between its vertices.
I
Which other graphs admit perfect state transfer?
Gabriel Coutinho
Algebraic graph theory
29 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Perfect state transfer
A graph X admits perfect state transfer between vertices u and v
if there is a time t such that:
exp(−ıtA(X ))eu = λev .
As we saw, K2 admits perfect state transfer between its vertices.
I
Which other graphs admit perfect state transfer?
Using the spectral decomposition of the past slide, one can find
that P3 admits perfect state transfer between its end vertices at
time π.
Gabriel Coutinho
Algebraic graph theory
29 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Perfect state transfer
A graph X admits perfect state transfer between vertices u and v
if there is a time t such that:
exp(−ıtA(X ))eu = λev .
As we saw, K2 admits perfect state transfer between its vertices.
I
Which other graphs admit perfect state transfer?
Using the spectral decomposition of the past slide, one can find
that P3 admits perfect state transfer between its end vertices at
time π.
But no other path does.
Gabriel Coutinho
Algebraic graph theory
29 / 30
What is it about? - the tools
Problem 1 - (very) regular graphs
Problem 2 - quantum walks
Definition
Spectral decomposition
Recent results and open problems
Recently I’ve been investigating classes of graphs that admit
perfect state transfer.
I
Problem is solved for distance-regular graphs (generalization
of strongly regular graphs with larger diameter) - joint work
with C. Godsil, K. Guo and F. Vanhove.
I
Graphs which are products of other graphs - joint work with
C. Godsil.
Questions:
I
Which graph properties can be derived from properties of
exp(−ıtA) ?
I
More classes of graphs admitting perfect state transfer specially graphs with few edges.
Gabriel Coutinho
Algebraic graph theory
30 / 30