An Introduction to Spectral Graph Theory

An Introduction to Spectral Graph Theory Mackenzie Wheeler Supervisor: Dr. Gary MacGillivray University of Victoria WheelerM@uvic.ca

Outline

Outline 1. How many walks are there from vertices v i to v j of length d?

Outline 1. How many walks are there from vertices v i to v j of length d? 2. How many labelled spanning trees of G exist?

Outline 1. How many walks are there from vertices v i to v j of length d? 2. How many labelled spanning trees of G exist? Graph Theory Review Define the Adjacency matrix A(G) Answer Question 1 Linear Algebra Review Define the Laplacian matrix L(G) Answer Question 2

Graph Theory Review Definition Two vertices v i and v j V (G) are said to be adjacent if {v i, v j } E(G).

Graph Theory Review Definition Two vertices v i and v j V (G) are said to be adjacent if {v i, v j } E(G). v 1 v 5 v 2 v 4 v 3

Graph Theory Review Definition Two vertices v i and v j V (G) are said to be adjacent if {v i, v j } E(G). v 1 v 5 v 2 v 4 v 3 Definition A walk in a graph G is a sequence of vertices {v 1, v 2,..., v k } such that v i is adjacent to v i+1 for all 1 i k 1. The length of the walk is k 1.

The Adjacency Matrix Definition The adjacency matrix A(G) of a graph G is defined by (A(G)) ij = { 1 v i v j E(G) 0 v i v j / E(G)

The Adjacency Matrix Definition The adjacency matrix A(G) of a graph G is defined by (A(G)) ij = { 1 v i v j E(G) 0 v i v j / E(G) Example Let G = C 5, then we have that 0 1 0 0 1 1 0 1 0 0 A(C 5 ) = 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0

Counting walks Question Given a graph G, how many walks are there from v i to v j of length d?

Counting walks Question Given a graph G, how many walks are there from v i to v j of length d? Proposition Let G be a graph with n vertices and adjacency matrix A(G), then the number of walks from v i to v j of length d in G is given by (A(G)) d ij.

The Adjacency Matrix Proof.

The Adjacency Matrix Proof. For d = 1, A d is A.

The Adjacency Matrix Proof. For d = 1, A d is A. Consider A d+1 = A d A

The Adjacency Matrix Proof. For d = 1, A d is A. Consider A d+1 = A d A Then a d+1 ij = n k=1 ad ik a kj.

The Adjacency Matrix Proof. For d = 1, A d is A. Consider A d+1 = A d A Then a d+1 ij = n k=1 ad ik a kj. a d ik a kj is the number of walks from v i to v j which are walks from v i to v k of length d, followed by a walk of length 1 from from v k to v j.

The Adjacency Matrix Proof. For d = 1, A d is A. Consider A d+1 = A d A Then a d+1 ij = n k=1 ad ik a kj. a d ik a kj is the number of walks from v i to v j which are walks from v i to v k of length d, followed by a walk of length 1 from from v k to v j. Therefore a n+1 ij = n k=1 ad ik a kj is to total number of walks from v i to v j of length d + 1.

The Adjacency Matrix Corollary Let G be a graph with e edges and t triangles, then 1. tr(a(g) 2 ) = 2e 2. tr(a(g) 3 ) = 6t

The Laplacian Matrix Definition The Laplacian matrix L(G) of a graph G is defined by deg(v i ) i = j (L(G)) ij = 1 i j and v i v j E(G) 0 otherwise

The Laplacian Matrix Definition The Laplacian matrix L(G) of a graph G is defined by deg(v i ) i = j (L(G)) ij = 1 i j and v i v j E(G) 0 otherwise Example Let G = C 5, then we have that 2 1 0 0 1 1 2 1 0 0 L(C 5 ) = 0 1 2 1 0 0 0 1 2 1 1 0 0 1 2

Linear Algebra Review Definition Let A M n n (R) and let v R n be a nonzero vector. Then v is an eigenvector of A if there exists a scalar λ R, such that Av = λv. We say that λ is an eigenvalue of A with corresponding eigenvector v.

Linear Algebra Review Definition Let A M n n (R) and let v R n be a nonzero vector. Then v is an eigenvector of A if there exists a scalar λ R, such that Av = λv. We say that λ is an eigenvalue of A with corresponding eigenvector v. Proposition Let A M n n (R) be a symmetric matrix, then the eigenvalues of A are all real numbers.

Linear Algebra Review Definition Let A M n n (R), and let a ij denote the entry in the i th row and j th column. A is diagonally dominant if a ii j i a ij for all 1 i n.

Linear Algebra Review Definition Let A M n n (R), and let a ij denote the entry in the i th row and j th column. A is diagonally dominant if a ii j i a ij for all 1 i n. Example 7 1 0 2 1 1 6 1 0 0 A = 0 1 3 1 0 0 0 1 2 1 2 0 0 1 4

Linear Algebra Review Proposition Let A be a symmetric, diagonally dominant n n matrix such that a ii > 0 for all a i n. Then all the eigenvalues of A are non-negative.

Linear Algebra Review Proposition Let A be a symmetric, diagonally dominant n n matrix such that a ii > 0 for all a i n. Then all the eigenvalues of A are non-negative. Corollary Let G be a graph with Laplacian L(G). The eigenvalues of L(G) are all nonnegative real numbers. Therefore, we may list the eigenvalues of L(G) as 0 = λ 0 λ 1 λ 2 λ n 1.

The Laplacian Matrix

The Laplacian Matrix Proposition Let G be a graph with Laplacian matrix L(G). Then λ = 0 is an eigenvalue of L(G) with v = (1, 1,..., 1) as a corresponding eigenvector.

The Laplacian Matrix Proposition Let G be a graph with Laplacian matrix L(G). Then λ = 0 is an eigenvalue of L(G) with v = (1, 1,..., 1) as a corresponding eigenvector. Proposition Let G be a connected graph with Laplacian L(G). Then λ = 0 is an eigenvalue of L(G) with multiplicity one.

Counting Labelled Spanning Trees

Counting Labelled Spanning Trees Definition A graph G is a tree if G is connected and contains no cycles.

Counting Labelled Spanning Trees Definition A graph G is a tree if G is connected and contains no cycles. Definition Let G be a graph with a subgraph T. T is a spanning tree of G if V (T ) = V (G) and T is a tree.

Counting Labelled Spanning Trees Definition A graph G is a tree if G is connected and contains no cycles. Definition Let G be a graph with a subgraph T. T is a spanning tree of G if V (T ) = V (G) and T is a tree. Example The Petersen Graph

Counting Labelled Spanning Trees Definition A graph G is a tree if G is connected and contains no cycles. Definition Let G be a graph with a subgraph T. T is a spanning tree of G if V (T ) = V (G) and T is a tree. Example A spanning tree of the Petersen graph

Counting Labelled Spanning Trees Question Given a graph G with vertices labelled {v 1, v 2,..., v n } how many labelled spanning trees of G exist?

Counting Labelled Spanning Trees Question Given a graph G with vertices labelled {v 1, v 2,..., v n } how many labelled spanning trees of G exist? Theorem (Kirchoff s Theorem) Let G be a connected graph with n 2 labelled vertices, and let 0 = λ 0 < λ 1 λ 2 λ n 1 be the eigenvalues of L(G). Then the number of spanning trees on G, t(g), is given by n 1 t(g) = det(l(g)[i]) = 1 n k=1. Where L(G)[i] denotes the matrix obtained from L(G) by deleteing the i th row and i th column. λ k

Counting Labelled Spanning Trees Proof Outline:

Counting Labelled Spanning Trees Proof Outline: We preceed by induction on V (G) + E(G) = n + m

Counting Labelled Spanning Trees Proof Outline: We preceed by induction on V (G) + E(G) = n + m When n + m = 3, the only connected graph is P 2

Counting Labelled Spanning Trees Proof Outline: We preceed by induction on V (G) + E(G) = n + m When n + m = 3, the only connected graph is P 2 P 2 L(P 2 ) = [ 1 ] 1 1 1

Counting Labelled Spanning Trees Proof Outline:

Counting Labelled Spanning Trees Proof Outline: Consider a graph V (G) + E(G) = n + m + 1

Counting Labelled Spanning Trees Proof Outline: Consider a graph V (G) + E(G) = n + m + 1 Let e = v i v j be an edge incident with the vertex v i

Counting Labelled Spanning Trees Proof Outline: Consider a graph V (G) + E(G) = n + m + 1 Let e = v i v j be an edge incident with the vertex v i Notice that t(g) = t(g e) + t(g \ e)

Counting Labelled Spanning Trees Proof Outline: Consider a graph V (G) + E(G) = n + m + 1 Let e = v i v j be an edge incident with the vertex v i Notice that t(g) = t(g e) + t(g \ e) v 1 v 2 v 2 v 4 v 3 (v 1, v 4 ) v 3

Counting Labelled Spanning Trees Proof Outline: Consider a graph V (G) + E(G) = n + m + 1 Let e = v i v j be an edge incident with the vertex v i Notice that t(g) = t(g e) + t(g \ e) v 1 v 2 v 2 v 4 v 3 (v 1, v 4 ) 2 2 0 L(G \ e) = 2 3 1 0 1 1 v 3

Counting Labelled Spanning Trees Proof Outline: By standard manipulation of the determinant we get det(l(g)[i]) = det(l(g e)[i]) + det(l(g \ e)[j]) = t(g e) + t(g \ e), by induction hypothesis = t(g).

Counting Labelled Spanning Trees Corollary (Cayley s Formula) The number of labelled spanning trees on the complete graph K n is n n 2.

Counting Labelled Spanning Trees Corollary (Cayley s Formula) The number of labelled spanning trees on the complete graph K n is n n 2. Proof. The eigenvalues of L(K n ) are 0 and n with multiplicity 1 and n 1, respectively. Therefore, by Kirchoff s Theorem the number of spanning trees on K n is nn 1 n = n n 2.

References C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, 2001. A. E. Brouwer and W..H Haemers, Spectra of Graphs, Springer, 2011.