An Introduction to Algebraic Graph Theory Rob Beezer [email protected] Department of Mathematics and Computer Science University of Puget Sound Mathematics Department Seminar Pacific University October 19, 2009 Labeling Puzzles −1 1 Assign a single real number value to each circle. For each circle, sum the 0 0 values of adjacent circles. Goal: Sum at each circle should be a common multiple of the 1 −1 value at the circle. Common multiple: −2 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 2 / 36 Example Solutions 1 1 1 1 1 1 Common multiple: 3 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 3 / 36 Example Solutions 1 −1 1 −1 1 −1 Common multiple: 1 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 4 / 36 Example Solutions −1 −1 0 0 1 1 Common multiple: 0 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 5 / 36 Example Solutions 0 −1 1 0 Common multiple: −1 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 6 / 36 Example Solutions −1 0 0 1 Common multiple: 0 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 7 / 36 Example Solutions r−1 4 r=Sqrt(17) 4 r−1 p 1 1 Common multiple: 2 (r + 1) = 2 17 + 1 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 8 / 36 Graphs A graph is a collection of vertices (nodes, dots) where some pairs are joined by edges (arcs, lines). The geometry of the vertex placement, or the contours of the edges are irrelevant. The relationships between vertices are important. Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 9 / 36 u v Always a symmetric matrix with zero diagonal. Useful for computer representations. x w Entr´eeto linear algebra, especially u v w x eigenvalues and eigenvectors. u 0 1 0 1 Symmetry groups of graphs is the other v 1 0 1 1 branch of Algebraic Graph Theory. w 0 1 0 1 x 1 1 1 0 Adjacency Matrix Given a graph, build a matrix of zeros and ones as follows: Label rows and columns with vertices, in the same order. Put a 1 in an entry if the corresponding vertices are connected by an edge. Otherwise put a 0 in the entry. Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 10 / 36 u v Always a symmetric matrix with zero diagonal. Useful for computer representations. x w Entr´eeto linear algebra, especially eigenvalues and eigenvectors. Symmetry groups of graphs is the other branch of Algebraic Graph Theory. Adjacency Matrix Given a graph, build a matrix of zeros and ones as follows: Label rows and columns with vertices, in the same order. Put a 1 in an entry if the corresponding vertices are connected by an edge. Otherwise put a 0 in the entry. u v w x u 0 1 0 1 v 1 0 1 1 w 0 1 0 1 x 1 1 1 0 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 10 / 36 u v x w Adjacency Matrix Given a graph, build a matrix of zeros and ones as follows: Label rows and columns with vertices, in the same order. Put a 1 in an entry if the corresponding vertices are connected by an edge. Otherwise put a 0 in the entry. Always a symmetric matrix with zero diagonal. Useful for computer representations. Entr´eeto linear algebra, especially u v w x eigenvalues and eigenvectors. u 0 1 0 1 Symmetry groups of graphs is the other v 1 0 1 1 branch of Algebraic Graph Theory. w 0 1 0 1 x 1 1 1 0 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 10 / 36 Adjacency matrix is real, symmetric ) real eigenvalues, algebraic and geometric multiplicities are equal minimal polynomial is product of linear factors for distinct eigenvalues Eigenvalues: λ = 3 m = 1 λ = 1 m = 5 λ = −2 m = 4 Eigenvalues of Graphs λ is an eigenvalue of a graph , λ is an eigenvalue of the adjacency matrix , A~x = λ~x for some vector ~x Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 11 / 36 Eigenvalues: λ = 3 m = 1 λ = 1 m = 5 λ = −2 m = 4 Eigenvalues of Graphs λ is an eigenvalue of a graph , λ is an eigenvalue of the adjacency matrix , A~x = λ~x for some vector ~x Adjacency matrix is real, symmetric ) real eigenvalues, algebraic and geometric multiplicities are equal minimal polynomial is product of linear factors for distinct eigenvalues Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 11 / 36 Eigenvalues of Graphs λ is an eigenvalue of a graph , λ is an eigenvalue of the adjacency matrix , A~x = λ~x for some vector ~x Adjacency matrix is real, symmetric ) real eigenvalues, algebraic and geometric multiplicities are equal minimal polynomial is product of linear factors for distinct eigenvalues Eigenvalues: λ = 3 m = 1 λ = 1 m = 5 λ = −2 m = 4 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 11 / 36 4-Dimensional Cube 0011 0111 0001 0101 0010 Vertices: Length 4 binary strings 1011 Join strings differing in exactly one bit 0000 1111 Generalizes 3-D cube 0100 1101 1010 1000 1110 1100 Eigenvalues: λ = 4 λ = 2 λ = 0 λ = −2 λ = −4 m = 1 m = 4 m = 6 m = 4 m = 1 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 12 / 36 Regular Graphs A graph is regular if every vertex has the same number of edges incident. The degree is the common number of incident edges. Theorem Suppose G is a regular graph of degree r. Then r is an eigenvalue of G The multiplicity of r is the number of connected components of G Regular of degree 3 with 2 components implies that λ = 3 will be an eigenvalue of multiplicity 2. Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 13 / 36 Proof. Let ~u be the vector where every entry is 1. Then 213 2r3 617 6r7 A~u = A 6 7 = 6 7 = r~u 6.7 6.7 4.5 4.5 1 r For each component of the graph, form a vector with 1's in entries corresponding to the vertices of the component, and zeros elsewhere. These eigenvectors form a basis for the eigenspace of r. (Their sum is the vector ~u above.) Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 14 / 36 Labeling Puzzles Explained The product of a graph's adjacency matrix with a column vector, A~u, forms sums of entries of ~u for all adjacent vertices. If ~u is an eigenvector, then these sums should equal a common multiple of the numbers assigned to each vertex. This multiple is the eigenvalue. So the puzzles earlier were simply asking for: eigenvectors (assignments of numbers to vertices), and eigenvalues (common multiples) of the adjacency matrix of the graph. Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 15 / 36 Eigenvalues and Eigenvectors of the Prism 20 1 0 1 0 13 2 3 61 0 1 0 0 17 6 7 60 1 0 1 1 07 1 A = 6 7 61 0 1 0 1 07 6 7 4 40 0 1 1 0 15 1 1 0 0 1 0 6 5 λ = 3 λ = 1 λ = 0 λ = −2 213 2 1 3 2 0 3 2 1 3 2 0 3 2−13 617 6 1 7 6−17 6−17 6−17 6 1 7 6 7 6 7 6 7 6 7 6 7 6 7 617 6−17 6−17 6−17 6 1 7 6−17 6 7 6 7 6 7 ; 6 7 6 7 ; 6 7 617 6−17 6 0 7 6 1 7 6 0 7 6 1 7 6 7 6 7 6 7 6 7 6 7 6 7 415 4−15 4 1 5 4 0 5 4−15 4 0 5 1 1 1 0 1 0 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 16 / 36 Eigenvalues and Eigenvectors 1 2 20 1 0 13 61 0 1 17 A = 6 7 40 1 0 15 1 1 1 0 3 4 1 p 1 p λ = −1 λ = 0 λ = 17 + 1 λ = − 17 + 1 p2 p2 2 0 3 2−13 2 17 − 13 2 17 + 13 6−17 6 0 7 6 4 7 6 −4 7 6 7 6 7 6p 7 6p 7 4 0 5 4 1 5 4 17 − 15 4 17 + 15 1 0 4 −4 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 17 / 36 Proof. Base case: ` = 1. Adjacency matrix describes walks of length 1. n h `+1i h ` i X h `i X h `i A = A A = A [A]kj = A ij ij ik ik k=1 k:vk adj vj = total ways to walk in ` steps from vi to vk , a neighbor of vj l k i k j l Powers of Adjacency Matrices Theorem Suppose A is the adjacency matrix of a graph.