In which we review linear algebra and introduce spectral graph theory.
1. Eigenvalues and Eigenvectors
Spectral graph theory studies how the eigenvalues of the adjacency matrix of a graph, which are purely algebraic quantities, relate to combinatorial properties of the graph.
We begin with a brief review of linear algebra.
If is a complex number, then we let denote its conjugate.
If is a square matrix, is a scalar, is a non-zero vector and we have
then we say that is an eigenvalue of and that is eigenvector of corresponding to the eigenvalue .
When (1) is satisfied, then we equivalently have
for a non-zero vector , which is equivalent to
For a fixed matrix , the function is a univariate polynomial of degree in and so, over the complex numbers, the equation (2) has exactly solutions, counting multiplicities.
If is a graph, then we will be interested in the adjacency matrix of , that is the matrix such that if and otherwise. If is a multigraph or a weighted graph, then is equal to the number of edges between , or the weight of the edge , respectively.
The adjacency matrix of an undirected graph is symmetric, and this implies that its eigenvalues are all real.
Definition 1 A matrix is Hermitian if for every .
Note that a real symmetric matrix is always Hermitian.
Lemma 2 If is Hermitian, then all the eigenvalues of are real.
Proof: Let be an Hermitian matrix and let be a scalar and be a non-zero vector such that . We will show that , which implies that is a real number. We define the following inner product operation over vectors in :
Notice that, by definition, we have and . The lemma follows by observing that
where we use the fact that is Hermitian, and that
so that .
From the discussion so far, we have that if is the adjacency matrix of an undirected graph then it has real eigenvalues, counting multiplicities of the number of solutions to .
If is a -regular graph, then instead of working with the adjacency matrix of it is somewhat more convenient to work with the normalized matrix .
In the rest of this section we shall prove the following relations between the eigenvalues of and certain purely combinatorial properties of .
Theorem 3 Let be a -regular undirected graph, and be its normalized adjacency matrix. Let be the real eigenvalues of with multiplicities. Then
- and .
- if and only if is disconnected.
- if and only if at least one of the connected components of is bipartite.
In the next lecture we will begin to explore an “approximate” version of the second claim, and to show that is close to 1 if and only if has a sparse cut.