Algebraic graph theory Gabriel Coutinho University of Waterloo November 6th, 2013 We can associate many matrices to a graph X . Adjacency: 0 0 1 0 1 0 1 B 1 0 1 0 1 C B C A(X ) = B 0 1 0 1 1 C B C @ 1 0 1 0 0 A 0 1 1 0 0 Tutte: 0 1 1 0 0 0 0 1 0 1 0 x12 0 x14 0 1 0 1 1 0 0 B C B −x12 0 x23 0 x25 C Incidence: B 0 0 1 0 1 1 C B C B C B 0 −x23 0 x34 x35 C B 0 1 0 0 0 1 C B C @ A @ −x14 0 −x34 0 0 A 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 0 1 0 1 0 1 B 1 0 1 0 1 C B C A(X ) = B 0 1 0 1 1 C B C @ 1 0 1 0 0 A 0 1 1 0 0 Tutte: 0 1 1 0 0 0 0 1 0 1 0 x12 0 x14 0 1 0 1 1 0 0 B C B −x12 0 x23 0 x25 C Incidence: B 0 0 1 0 1 1 C B C B C B 0 −x23 0 x34 x35 C B 0 1 0 0 0 1 C B C @ A @ −x14 0 −x34 0 0 A 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 0 1 0 1 0 1 B 1 0 1 0 1 C B C A(X ) = B 0 1 0 1 1 C B C @ 1 0 1 0 0 A 0 1 1 0 0 Tutte: 0 1 1 0 0 0 0 1 0 1 0 x12 0 x14 0 1 0 1 1 0 0 B C B −x12 0 x23 0 x25 C Incidence: B 0 0 1 0 1 1 C B C B C B 0 −x23 0 x34 x35 C B 0 1 0 0 0 1 C B C @ A @ −x14 0 −x34 0 0 A 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: 0 1 1 0 0 0 0 1 0 1 0 x12 0 x14 0 1 0 1 1 0 0 B C B −x12 0 x23 0 x25 C Incidence: B 0 0 1 0 1 1 C B C B C B 0 −x23 0 x34 x35 C B 0 1 0 0 0 1 C B C @ A @ −x14 0 −x34 0 0 A 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 0 1 0 1 0 1 B 1 0 1 0 1 C B C A(X ) = B 0 1 0 1 1 C B C @ 1 0 1 0 0 A 0 1 1 0 0 Gabriel Coutinho Algebraic graph theory 2 / 30 Tutte: 0 1 0 x12 0 x14 0 B −x12 0 x23 0 x25 C B C B 0 −x23 0 x34 x35 C B C @ −x14 0 −x34 0 0 A 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 0 1 0 1 0 1 B 1 0 1 0 1 C B C A(X ) = B 0 1 0 1 1 C B C @ 1 0 1 0 0 A 0 1 1 0 0 0 1 1 0 0 0 0 1 B 1 0 1 1 0 0 C B C Incidence: B 0 0 1 0 1 1 C B C @ 0 1 0 0 0 1 A 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 0 1 0 1 0 1 B 1 0 1 0 1 C B C A(X ) = B 0 1 0 1 1 C B C @ 1 0 1 0 0 A 0 1 1 0 0 Tutte: 0 1 1 0 0 0 0 1 0 1 0 x12 0 x14 0 1 0 1 1 0 0 B C B −x12 0 x23 0 x25 C Incidence: B 0 0 1 0 1 1 C B C B C B 0 −x23 0 x34 x35 C B 0 1 0 0 0 1 C B C @ A @ −x14 0 −x34 0 0 A 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 0 1 0 1 0 1 B 1 0 1 0 1 C B C Adjacency: B 0 1 0 1 1 C B C @ 1 0 1 0 0 A 0 1 1 0 0 Tutte: 0 1 1 0 0 0 0 1 0 1 0 x12 0 x14 0 1 0 1 1 0 0 B C B −x12 0 x23 0 x25 C Incidence: B 0 0 1 0 1 1 C B C B C B 0 −x23 0 x34 x35 C B 0 1 0 0 0 1 C B C @ A @ −x14 0 −x34 0 0 A 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.