Honors 1: Undergraduate Math Lab^{∗}

Peter Sarnak^{†}, Steven Miller^{‡}, Alex Barnett^{§}

September 25th, 2002

Courant Institute of Mathematical Sciences New York University New York, NY

Abstract

We review some basic definitions of Graph Theory, and prove a simple result about the size of the eigenvalues of adjacency graphs. Lecture by Peter Sarnak; notes by Steven Miller.

1 Definitions

Definition 1.1 For a graph G, let V (G) be the set of vertices and E(G) the set of edges (an edge is a pair (v, w) with v, w ∈ V). We often just write V and E.

Definition 1.2 (Connected Graph) A graph G is connected if for any two vertices v, w ∈ G, there is a path of edges in E starting at v and ending at w.

Definition 1.3 (Boundary) ∂A = {v ∈ V − A : there is a w ∈ A with (v, w) ∈ E}.

∗Homepage: http://www.math.nyu.edu/∼millerj/

†E-mail: sarnak@math.princeton.edu

‡E-mail: millerj@cims.nyu.edu or sjmiller@math.princeton.edu

§E-mail: barnett@nmr.mgh.harvard.edu

Definition 1.4 (k-regular) A graph G is k-regular if there are k edges coming out from each vertex.

Definition 1.5 (Expander Graph) Let G be a connected graph with n vertices. Let A ⊂ V (G) be any subset of vertices with |A| ≤ \frac{n}{2} (ie, at most half of the vertices). We say G is a (n, c, k) expander if for any such A, |∂A| ≥ c|A|.

Example 1.6 Let G be a 2-regular graph with n vertices. For definateness, label the vertices {1, 2, …, n}. Let the edges be (1, 2), (2, 3), (3, 4), …, (n, 1). Let A be the first half of the vertices: A = \{1, 2, …, [\frac{n}{2}]\}. Then |∂A| = 2, and G is not an (n, c, 2) expander.

2 Size of Eigenvalues

Consider a 3-regular graph with n vertices. We want the graph to have certain desirable connectivity properties.

Let A = (a_{vw}) is the adjacency matrix attached to the graph G, and a_{vw} = 1 if there is an edge from v to w and 0 otherwise. This is a very sparse matrix (only three non-zero entries in each row or column). Compute its eigenvalues (real numbers).

Lemma 2.1 The eigenvalues λ_{i} ∈ [−3, 3].

Proof: Let f : V → ℝ, define the action of the adjacency matrix A on f by

Af(v) = ∑_{(v,w)∈E} f(w).  (1)

As there are finitely many vertices (n, in fact), we can regard the function f(v) as living in ℝ^{n}, with coordinates (f(v_{1}), …, f(v_{n})).

Suppose ∀w ∈ V, Af(w) = λf(w), f ≠ 0. Then λ is an eigenvalue (which must be real as A is real symmetric).

We use the Maximum Modulus Principle. As f ≠ 0, let w_{0} be such that f(w_{0}) is the maximum value of f(w) (not zero, and exists as we have finitely many vertices). Then

f(w_{0}) = \frac{1}{λ} ∑_{(w,w_{0})} f(w).  (2)

If λ > 3, this cannot happen (we’re assuming the graph is k-regular and connected). If λ = 3, then f is constant. Working with absolute values, we similarly obtain λ > −3.

Exercise 2.2 Prove a k-regular graph G is connected if and only if λ = k is a simple eigenvalue.

Let λ_{1} be the second largest eigenvalue; λ = k is always an eigenvalue for a k-regular connected graph. How big can the gap be between λ_{1} and k?