Algebraic graph theory
Gabriel Coutinho
University of Waterloo
November 6th, 2013 We can associate many matrices to a graph X .
Adjacency: 0 1 0 1 0 1 0 1 0 1 A(X ) = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 Tutte: 1 1 0 0 0 0 0 x12 0 x14 0 1 0 1 1 0 0 −x12 0 x23 0 x25 Incidence: 0 0 1 0 1 1 0 −x23 0 x34 x35 0 1 0 0 0 1 −x14 0 −x34 0 0 0 0 0 1 1 0 0 −x25 −x35 0 0
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
Gabriel Coutinho Algebraic graph theory 2 / 30 Adjacency: 0 1 0 1 0 1 0 1 0 1 A(X ) = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 Tutte: 1 1 0 0 0 0 0 x12 0 x14 0 1 0 1 1 0 0 −x12 0 x23 0 x25 Incidence: 0 0 1 0 1 1 0 −x23 0 x34 x35 0 1 0 0 0 1 −x14 0 −x34 0 0 0 0 0 1 1 0 0 −x25 −x35 0 0
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
We can associate many matrices to a graph X .
Gabriel Coutinho Algebraic graph theory 2 / 30 Adjacency: 0 1 0 1 0 1 0 1 0 1 A(X ) = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 Tutte: 1 1 0 0 0 0 0 x12 0 x14 0 1 0 1 1 0 0 −x12 0 x23 0 x25 Incidence: 0 0 1 0 1 1 0 −x23 0 x34 x35 0 1 0 0 0 1 −x14 0 −x34 0 0 0 0 0 1 1 0 0 −x25 −x35 0 0
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
We can associate many matrices to a graph X .
Gabriel Coutinho Algebraic graph theory 2 / 30 Tutte: 1 1 0 0 0 0 0 x12 0 x14 0 1 0 1 1 0 0 −x12 0 x23 0 x25 Incidence: 0 0 1 0 1 1 0 −x23 0 x34 x35 0 1 0 0 0 1 −x14 0 −x34 0 0 0 0 0 1 1 0 0 −x25 −x35 0 0
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
We can associate many matrices to a graph X .
Adjacency: 0 1 0 1 0 1 0 1 0 1 A(X ) = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0
Gabriel Coutinho Algebraic graph theory 2 / 30 Tutte: 0 x12 0 x14 0 −x12 0 x23 0 x25 0 −x23 0 x34 x35 −x14 0 −x34 0 0 0 −x25 −x35 0 0
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
We can associate many matrices to a graph X .
Adjacency: 0 1 0 1 0 1 0 1 0 1 A(X ) = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 Incidence: 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 1 0
Gabriel Coutinho Algebraic graph theory 2 / 30 What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
We can associate many matrices to a graph X .
Adjacency: 0 1 0 1 0 1 0 1 0 1 A(X ) = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 Tutte: 1 1 0 0 0 0 0 x12 0 x14 0 1 0 1 1 0 0 −x12 0 x23 0 x25 Incidence: 0 0 1 0 1 1 0 −x23 0 x34 x35 0 1 0 0 0 1 −x14 0 −x34 0 0 0 0 0 1 1 0 0 −x25 −x35 0 0
Gabriel Coutinho Algebraic graph theory 2 / 30 I Implies that LPs with coefficient matrix D are integral.
I Can be used to prove max-flow-min-cut theorem and other results. T I The matrix Q = DD 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 .
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Properties - incidence matrix
I The incidence matrix D of directed graphs is totally unimodular.
Gabriel Coutinho Algebraic graph theory 3 / 30 I Can be used to prove max-flow-min-cut theorem and other results. T I The matrix Q = DD 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 .
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 T I The matrix Q = DD 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 .
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 I Using linear algebra, one can prove that the determinant of a submatrix of Q counts the number of spanning trees of X .
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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. T I The matrix Q = DD is called Laplacian of the underlying undirected graph X .
Gabriel Coutinho Algebraic graph theory 3 / 30 What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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. T I The matrix Q = DD 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
0 1 0 1 0 1 0 1 0 1 Adjacency: 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 Tutte: 1 1 0 0 0 0 0 x12 0 x14 0 1 0 1 1 0 0 −x12 0 x23 0 x25 Incidence: 0 0 1 0 1 1 0 −x23 0 x34 x35 0 1 0 0 0 1 −x14 0 −x34 0 0 0 0 0 1 1 0 0 −x25 −x35 0 0
Gabriel Coutinho Algebraic graph theory 4 / 30 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Properties - Tutte matrix
I It’s determinant is a formal expression on those variables.
Gabriel Coutinho Algebraic graph theory 5 / 30 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 I Can be used to provide state of the art algorithms to find matchings.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Matrices
0 1 0 1 0 1 0 1 0 1 Adjacency: 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 Tutte: 1 1 0 0 0 0 0 x12 0 x14 0 1 0 1 1 0 0 −x12 0 x23 0 x25 Incidence: 0 0 1 0 1 1 0 −x23 0 x34 x35 0 1 0 0 0 1 −x14 0 −x34 0 0 0 0 0 1 1 0 0 −x25 −x35 0 0
Gabriel Coutinho Algebraic graph theory 6 / 30 0 1 0 1 0 1 0 1 0 1 A = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 2 0 2 0 1 0 5 1 4 2 0 3 1 2 1 5 2 6 1 4 A2 = 2 1 3 0 1 A3 = 1 6 2 5 4 0 2 0 2 1 4 1 5 0 2 1 1 1 1 2 2 4 4 2 2 Can model random walks. Second largest eigenvalue determines how fast a random walk becomes really random. Applications in cryptography.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Properties - Adjacency matrix A
Entries of Ak count the number of walks of length k.
Gabriel Coutinho Algebraic graph theory 7 / 30 2 0 2 0 1 0 5 1 4 2 0 3 1 2 1 5 2 6 1 4 A2 = 2 1 3 0 1 A3 = 1 6 2 5 4 0 2 0 2 1 4 1 5 0 2 1 1 1 1 2 2 4 4 2 2 Can model random walks. Second largest eigenvalue determines how fast a random walk becomes really random. Applications in cryptography.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Properties - Adjacency matrix A
Entries of Ak count the number of walks of length k. 0 1 0 1 0 1 0 1 0 1 A = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0
Gabriel Coutinho Algebraic graph theory 7 / 30 Can model random walks. Second largest eigenvalue determines how fast a random walk becomes really random. Applications in cryptography.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Properties - Adjacency matrix A
Entries of Ak count the number of walks of length k. 0 1 0 1 0 1 0 1 0 1 A = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 2 0 2 0 1 0 5 1 4 2 0 3 1 2 1 5 2 6 1 4 A2 = 2 1 3 0 1 A3 = 1 6 2 5 4 0 2 0 2 1 4 1 5 0 2 1 1 1 1 2 2 4 4 2 2
Gabriel Coutinho Algebraic graph theory 7 / 30 What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Properties - Adjacency matrix A
Entries of Ak count the number of walks of length k. 0 1 0 1 0 1 0 1 0 1 A = 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 2 0 2 0 1 0 5 1 4 2 0 3 1 2 1 5 2 6 1 4 A2 = 2 1 3 0 1 A3 = 1 6 2 5 4 0 2 0 2 1 4 1 5 0 2 1 1 1 1 2 2 4 4 2 2 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 I Automorphisms of a graph forms a group, the Aut(X ).
In this example, Aut = D3 × Z2
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 In this example, Aut = D3 × Z2
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 In this example, Aut = D3 × Z2
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Symmetries of a graph 2
I Almost all graphs have a trivial automorphism group... (Erd˝os and R´enyi)
Gabriel Coutinho Algebraic graph theory 9 / 30 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Symmetries of a graph 2
I Almost all graphs have a trivial automorphism group... (Erd˝os and R´enyi)
I ... but every finite group can be realized as the automorphism group of a finite graph. (Frucht)
Gabriel Coutinho Algebraic graph theory 9 / 30 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Symmetries of a graph 2
I Almost all graphs have a trivial automorphism group... (Erd˝os and R´enyi)
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 I Using something called Burnside Lemma, we can (easily) find the answer: 1540.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Symmetries of a graph 2
I Almost all graphs have a trivial automorphism group... (Erd˝os and R´enyi)
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 I Using something called Burnside Lemma, we can (easily) find the answer: 1540.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Symmetries of a graph 2
I Almost all graphs have a trivial automorphism group... (Erd˝os and R´enyi)
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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Symmetries of a graph 2
I Almost all graphs have a trivial automorphism group... (Erd˝os and R´enyi)
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 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}.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 In this example, G is the Quaternion Group {±1, ±i, ±j, ±k} and S = {i, −i, j, −j}.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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 Lov´asz,but this has received more attention. It is solved for some classes of groups.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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 Lov´asz,but this has received more attention. It is solved for some classes of groups.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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 Lov´asz,but this has received more attention. It is solved for some classes of groups.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 I Is every Cayley Graph (apart from K2) Hamiltonian? I More general conjecture by Lov´asz,but this has received more attention. It is solved for some classes of groups.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Lov´asz,but this has received more attention. It is solved for some classes of groups.
Gabriel Coutinho Algebraic graph theory 11 / 30 I Characteristic polynomial of A(X ): φX (t) = det(tI − A(X )). k I Matching polynomial µX (t). Coefficient of x (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 )
i j and TX (x, y) = x y 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 k I Matching polynomial µX (t). Coefficient of x (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 )
i j and TX (x, y) = x y 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 I Tutte polynomial. Two variables. Defined for multigraphs recursively as:
TX = TX \e + TX /e for any e ∈ E(X )
i j and TX (x, y) = x y 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 )). k I Matching polynomial µX (t). Coefficient of x (up to signing) counts the number of matchings incident to n − k vertices.
Gabriel Coutinho Algebraic graph theory 12 / 30 As opposed to what happened with matrices, none of these polynomials actually determine the graph.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 )). k I Matching polynomial µX (t). Coefficient of x (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 )
i j and TX (x, y) = x y is X has i bridges, j loops, and no other edges.
Gabriel Coutinho Algebraic graph theory 12 / 30 What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 )). k I Matching polynomial µX (t). Coefficient of x (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 )
i j and TX (x, y) = x y 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 φ(t) = t5 − 4t3 + 2t µ(t) = t5 − 4t3 + 2t T (x, y) = x4
φ(t) = t5 − 4t3 µ(t) = t5 − 4t3 + 2t T (x, y) = y + x + x2 + x3
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Examples
φ(t) = t5 − 4t3 µ(t) = t5 − 4t3 T (x, y) = x4
Gabriel Coutinho Algebraic graph theory 13 / 30 φ(t) = t5 − 4t3 µ(t) = t5 − 4t3 + 2t T (x, y) = y + x + x2 + x3
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Examples
φ(t) = t5 − 4t3 φ(t) = t5 − 4t3 + 2t µ(t) = t5 − 4t3 µ(t) = t5 − 4t3 + 2t T (x, y) = x4 T (x, y) = x4
Gabriel Coutinho Algebraic graph theory 13 / 30 What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks Polynomials Examples
φ(t) = t5 − 4t3 φ(t) = t5 − 4t3 + 2t µ(t) = t5 − 4t3 µ(t) = t5 − 4t3 + 2t T (x, y) = x4 T (x, y) = x4
φ(t) = t5 − 4t3 µ(t) = t5 − 4t3 + 2t T (x, y) = y + x + x2 + x3
Gabriel Coutinho Algebraic graph theory 13 / 30 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 I So the conjectured needs to be proved “only” for graphs with similar such polynomials.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 I Strongly related to certain models of statistical physics.
What is it about? - the tools Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 Matrices Problem 1 - (very) regular graphs Groups Problem 2 - quantum walks 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 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.
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks More regular than just regular
Look to the following graph.
Gabriel Coutinho Algebraic graph theory 16 / 30 The second pair has a common a neighbour.
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 The second pair has a common a neighbour.
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 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)
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs
A graph X is called strongly regular with parameters (n, k, a, c) if:
Gabriel Coutinho Algebraic graph theory 17 / 30 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)
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 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)
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 (5,2,0,1) They arise naturally in the study of finite geometries, design theory, equiangular lines and group theory. (6,4,2,4)
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 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. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 2
Typical problem about strongly regular graphs:
Gabriel Coutinho Algebraic graph theory 18 / 30 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. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 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. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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
Gabriel Coutinho Algebraic graph theory 18 / 30 I Consider A = A(X ). Note that A(X ) = J − I − A.
I Recall that powers of A count walks between vertices. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 I Recall that powers of A count walks between vertices. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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. 2 I What is (A )u,u? Valency of the graph: k.
Gabriel Coutinho Algebraic graph theory 18 / 30 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a.
Gabriel Coutinho Algebraic graph theory 18 / 30 2 I So A = kI + aA + c(J − I − A).
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c.
Gabriel Coutinho Algebraic graph theory 18 / 30 What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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. 2 I What is (A )u,u? Valency of the graph: k. 2 I If u ∼ v, then (A )u,v = a. 2 I If u v, then (A )u,v = c. 2 I So A = kI + aA + c(J − I − A).
Gabriel Coutinho Algebraic graph theory 18 / 30 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) −→ I If x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
I Hence
A2−→x = kI −→x +aA−→x +c(J −I −A)−→x ⇒ λ2 = k +aλ−c −cλ
a − c ± p(c − a)2 + 4(k − c) I So λ = 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 3
2 I A = kI + aA + c(J − I − A).
Gabriel Coutinho Algebraic graph theory 19 / 30 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) −→ I If x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
I Hence
A2−→x = kI −→x +aA−→x +c(J −I −A)−→x ⇒ λ2 = k +aλ−c −cλ
a − c ± p(c − a)2 + 4(k − c) I So λ = 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 3
2 I A = kI + aA + c(J − I − A).
I Can you tell who are the eigenvalues of A?
Gabriel Coutinho Algebraic graph theory 19 / 30 I Every other eigenvector is orthogonal to 1 (A is symmetric and should remember this from your 1st linear algebra course) −→ I If x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
I Hence
A2−→x = kI −→x +aA−→x +c(J −I −A)−→x ⇒ λ2 = k +aλ−c −cλ
a − c ± p(c − a)2 + 4(k − c) I So λ = 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 3
2 I A = 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 −→ I If x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
I Hence
A2−→x = kI −→x +aA−→x +c(J −I −A)−→x ⇒ λ2 = k +aλ−c −cλ
a − c ± p(c − a)2 + 4(k − c) I So λ = 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 3
2 I A = 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 I Hence
A2−→x = kI −→x +aA−→x +c(J −I −A)−→x ⇒ λ2 = k +aλ−c −cλ
a − c ± p(c − a)2 + 4(k − c) I So λ = 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 3
2 I A = 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) −→ I If x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
Gabriel Coutinho Algebraic graph theory 19 / 30 What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 3
2 I A = 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) −→ I If x is eigenvector for eigenvalue λ 6= k, then Jx = 0.
I Hence
A2−→x = kI −→x +aA−→x +c(J −I −A)−→x ⇒ λ2 = k +aλ−c −cλ
a − c ± p(c − a)2 + 4(k − c) I So λ = 2
Gabriel Coutinho Algebraic graph theory 19 / 30 I 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 + τmτ = 0. I You can solve the system. For example, if (n, k, a, c) = (58, 30, 20, 10), you get:
3 √ m = 19 − 7 5 θ 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 4
a − c ± p(c − a)2 + 4(k − c) I So λ = . 2
Gabriel Coutinho Algebraic graph theory 20 / 30 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 + τmτ = 0. I You can solve the system. For example, if (n, k, a, c) = (58, 30, 20, 10), you get:
3 √ m = 19 − 7 5 θ 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 4
a − c ± p(c − a)2 + 4(k − c) I So λ = . 2 I So you have three eigenvalues: k, θ, and τ.
Gabriel Coutinho Algebraic graph theory 20 / 30 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 + τmτ = 0. I You can solve the system. For example, if (n, k, a, c) = (58, 30, 20, 10), you get:
3 √ m = 19 − 7 5 θ 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 4
a − c ± p(c − a)2 + 4(k − c) I So λ = . 2 I So you have three eigenvalues: k, θ, and τ.
I The sum of their multiplicities is the order of A, so n.
Gabriel Coutinho Algebraic graph theory 20 / 30 I Easy to see: mk = 1.
I So 1 + mθ + mτ = n and k + θmθ + τmτ = 0. I You can solve the system. For example, if (n, k, a, c) = (58, 30, 20, 10), you get:
3 √ m = 19 − 7 5 θ 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 4
a − c ± p(c − a)2 + 4(k − c) I So λ = . 2 I 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.
Gabriel Coutinho Algebraic graph theory 20 / 30 I So 1 + mθ + mτ = n and k + θmθ + τmτ = 0. I You can solve the system. For example, if (n, k, a, c) = (58, 30, 20, 10), you get:
3 √ m = 19 − 7 5 θ 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 4
a − c ± p(c − a)2 + 4(k − c) I So λ = . 2 I 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.
Gabriel Coutinho Algebraic graph theory 20 / 30 I You can solve the system. For example, if (n, k, a, c) = (58, 30, 20, 10), you get:
3 √ m = 19 − 7 5 θ 2
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 4
a − c ± p(c − a)2 + 4(k − c) I So λ = . 2 I 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 + τmτ = 0.
Gabriel Coutinho Algebraic graph theory 20 / 30 What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Strongly regular graphs 4
a − c ± p(c − a)2 + 4(k − c) I So λ = . 2 I 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 + τmτ = 0. I You can solve the system. For example, if (n, k, a, c) = (58, 30, 20, 10), you get:
3 √ m = 19 − 7 5 θ 2
Gabriel Coutinho Algebraic graph theory 20 / 30 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.
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 So let’s look to simpler cases. Suppose a = 0. I Means adjacent vertices share no neighbours in common. Aka: no triangles.
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 I Means adjacent vertices share no neighbours in common. Aka: no triangles.
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 (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!
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks Open cases
The list of feasible parameter sets up to 300 vertices is:
Gabriel Coutinho Algebraic graph theory 22 / 30 If you require c = 1, Hoffman-Singleton theorem says that other than those three, you can only have: (3250,57,0,1) - major problem!
What is it about? - the tools Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 Starting Problem 1 - (very) regular graphs Triangle free Problem 2 - quantum walks 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 0 1/3 0 1/2 0 0 1/3 1/2 0 1/3 0 1/2 1 0 0 1/3 0 1/2 1/2 0 = 1/3 1/2 0 1/3 0 0 0 0 0 1/3 1/3 0 0 0 1/3
Powers of the scaled matrix represent the probability distribution of a longer walk...
5/12 0 5/18 0 1/6 1/6 1/6 1/6 1/6 1/6 0 4/9 1/6 5/12 1/6 1/4 1/4 1/4 1/4 1/4 2 100 Ae = 5/12 1/6 4/9 0 1/6 Ae ≈ 1/4 1/4 1/4 1/4 1/4 0 5/18 0 5/12 1/6 1/6 1/6 1/6 1/6 1/6 1/6 1/9 1/9 1/6 1/3 1/6 1/6 1/6 1/6 1/6
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Random walks
If X is a graph with adjacency matrix A, we can scale the columns of A to Ae in such a way that Ae represents a probability distribution of a particle moving in the graph.
Gabriel Coutinho Algebraic graph theory 23 / 30 Powers of the scaled matrix represent the probability distribution of a longer walk...
5/12 0 5/18 0 1/6 1/6 1/6 1/6 1/6 1/6 0 4/9 1/6 5/12 1/6 1/4 1/4 1/4 1/4 1/4 2 100 Ae = 5/12 1/6 4/9 0 1/6 Ae ≈ 1/4 1/4 1/4 1/4 1/4 0 5/18 0 5/12 1/6 1/6 1/6 1/6 1/6 1/6 1/6 1/9 1/9 1/6 1/3 1/6 1/6 1/6 1/6 1/6
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Random walks
If X is a graph with adjacency matrix A, we can scale the columns of A to Ae in such a way that Ae represents a probability distribution of a particle moving in the graph.
0 1/3 0 1/2 0 0 1/3 1/2 0 1/3 0 1/2 1 0 0 1/3 0 1/2 1/2 0 = 1/3 1/2 0 1/3 0 0 0 0 0 1/3 1/3 0 0 0 1/3
Gabriel Coutinho Algebraic graph theory 23 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Random walks
If X is a graph with adjacency matrix A, we can scale the columns of A to Ae in such a way that Ae represents a probability distribution of a particle moving in the graph.
0 1/3 0 1/2 0 0 1/3 1/2 0 1/3 0 1/2 1 0 0 1/3 0 1/2 1/2 0 = 1/3 1/2 0 1/3 0 0 0 0 0 1/3 1/3 0 0 0 1/3
Powers of the scaled matrix represent the probability distribution of a longer walk...
5/12 0 5/18 0 1/6 1/6 1/6 1/6 1/6 1/6 0 4/9 1/6 5/12 1/6 1/4 1/4 1/4 1/4 1/4 2 100 Ae = 5/12 1/6 4/9 0 1/6 Ae ≈ 1/4 1/4 1/4 1/4 1/4 0 5/18 0 5/12 1/6 1/6 1/6 1/6 1/6 1/6 1/6 1/9 1/9 1/6 1/3 1/6 1/6 1/6 1/6 1/6
Gabriel Coutinho Algebraic graph theory 23 / 30 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...
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 I For example, the probability distribution does not converge.
I But at some particular time, it might get concentrated...
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 I But at some particular time, it might get concentrated...
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 I Quantum theory says that the transition matrix is:
t2A2 ıt3A3 t4A4 exp(−ıtA) = I + −ıtA − + + − ..... 2 3! 4!
I Question: how do you compute that?
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Quantum walks 2
I We use a continuous-time.
Gabriel Coutinho Algebraic graph theory 25 / 30 I Question: how do you compute that?
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Quantum walks 2
I We use a continuous-time.
I Quantum theory says that the transition matrix is:
t2A2 ıt3A3 t4A4 exp(−ıtA) = I + −ıtA − + + − ..... 2 3! 4!
Gabriel Coutinho Algebraic graph theory 25 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Quantum walks 2
I We use a continuous-time.
I Quantum theory says that the transition matrix is:
t2A2 ıt3A3 t4A4 exp(−ıtA) = I + −ıtA − + + − ..... 2 3! 4!
I Question: how do you compute that?
Gabriel Coutinho Algebraic graph theory 25 / 30 A very nice thing about this matrix is that:
Ak = A if k is odd and Ak = I if k is even.
Hence cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 1 0
Gabriel Coutinho Algebraic graph theory 26 / 30 Hence cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 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:
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −ı sin(t) cos(t)
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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 Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
π t = 6
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
π t = 4
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
π t = 3
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
π t = 2
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
π π t = 2 + 6
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
π π t = 2 + 4
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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:
π π t = 2 + 3
Gabriel Coutinho Algebraic graph theory 26 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems Example
0 1 Suppose your graph is K . Then A(K ) = . 2 2 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 cos(t) −ı sin(t) exp(−ıtA) = cos(t)I − ı sin(t)A = −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 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 d X A = θr Er r=0 2 P with Er = Er , Er Es = 0 and Er = I .
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 I Suppose the distinct eigenvalues of A are {θ0, ..., θd }.
I Let Ek be a projector for the eigenspace of θk . I Then d X A = θr Er r=0 2 P with Er = Er , Er Es = 0 and Er = I .
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 I Let Ek be a projector for the eigenspace of θk . I Then d X A = θr Er r=0 2 P with Er = Er , Er Es = 0 and Er = I .
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 I Then d X A = θr Er r=0 2 P with Er = Er , Er Es = 0 and Er = I .
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 d X A = θr Er r=0 2 P with Er = Er , Er Es = 0 and Er = I .
Gabriel Coutinho Algebraic graph theory 27 / 30 d X A = θr Er r=0 then d k X k A = θr Er r=0 And then it is not difficult to convince yourself that:
d X −ıtθr exp(−ıtA) = e Er r=0
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 then d k X k A = θr Er r=0 And then it is not difficult to convince yourself that:
d X −ıtθr exp(−ıtA) = e Er r=0
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems In general - main tool 2
A very good feature of the spectral decomposition is that if:
d X A = θr Er r=0
Gabriel Coutinho Algebraic graph theory 28 / 30 And then it is not difficult to convince yourself that:
d X −ıtθr exp(−ıtA) = e Er r=0
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems In general - main tool 2
A very good feature of the spectral decomposition is that if:
d X A = θr Er r=0 then d k X k A = θr Er r=0
Gabriel Coutinho Algebraic graph theory 28 / 30 What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks Recent results and open problems In general - main tool 2
A very good feature of the spectral decomposition is that if:
d X A = θr Er r=0 then d k X k A = θr Er r=0 And then it is not difficult to convince yourself that:
d X −ıtθr exp(−ıtA) = e Er r=0
Gabriel Coutinho Algebraic graph theory 28 / 30 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.
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 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.
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 But no other path does.
What is it about? - the tools Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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 Definition Problem 1 - (very) regular graphs Spectral decomposition Problem 2 - quantum walks 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