Great Ideas in Theoretical Computer Science Summer 2013 Lecture 7: Expander Graphs in Computer Science
Lecturer: Kurt Mehlhorn & He Sun
Over the past decades expanders have become one basic tool in various areas of computer science, including algorithm design, complexity theory, coding theory, and etc. Informally, expanders are regular graphs with low degree and high connectivity. We can use different ways to define expander graphs.
1. Combinatorically, expanders are highly connected graphs, and to disconnect a large part of the graph, one has to sever many edges.
2. Geometrically, every vertex set has a relatively large boundary.
3. From the Probabilistic view, expanders are graphs whose behavior is “like” ran- dom graphs.
4. Algebraically, expanders are the real-symmetric matrix whose first positive eigen- value of the Laplace operator is bounded away from zero.
1 Spectra of Graphs
Definition 1. Let G = (V,E) be an undirected graph with vertex set [n] , {1, . . . , n}. The adjacency matrix of G is an n by n matrix A given by ( 1 if i and j are adjacent A = i,j 0 otherwise
If G is a multi-graph, then Ai,j is the number of edges between vertex i and vertex j. By definition, the adjacency matrix of graphs has the following properties: (1) The sum of elements in every row/column equals the degree of the corresponding vertex. (2) If G is undirected, then A is symmetric.
Example. The adjacency matrix of a triangle is
0 1 1 1 0 1 1 1 0
Given a matrix A, a vector x 6= 0 is defined to be an eigenvector of A if and only if there is a λ ∈ C such that Ax = λx. In this case, λ is called an eigenvalue of A.
1 Definition 2 (graph spectrum). Let A be the adjacency matrix of an undirected graph G with n vertices. Then A has n real eigenvalues, denoted by λ1 ≥ · · · ≥ λn. These eigenvalues associated with their multiplicities compose the spectrum of G.
Here are some basic facts about the graph spectrum.
Lemma 3. Let G be any undirected simple graph with n vertices. Then Pn 1. i=1 λi = 0. Pn 2 Pn 2. i=1 λi = i=1 deg(i).
3. If λ1 = ··· = λn, then E[G] = ∅.
4. degavg ≤ λ1 ≤ degmax. p 5. degmax ≤ λ1 ≤ degmax. Proof. We only prove the first three items. (1) Since G does not have self-loops, all the diagonal elements of A are zero. By the Pn Pn definition of trace, we have ı=1 λi = tr(A) = i=1 Ai,i = 0. Pn 2 2 Pn 2 (2) By the properties of matrix trace, we have ı=1 λi = tr (A ) = i=1 Ai,i. Since 2 2 Ai,i is the degree of vertex i, tr(A ) equals the sum of all vertices’ degrees in G. Pn (3) Combing i=1 λi = 0 with λ1 = ··· = λn, we have λi = 0 for every vertex i. By item (2), we have deg(i) = 0 for any vertex i. Therefore E[G] = ∅.
k For a graph G with adjacency matrix A and integer k ≥ 1, Au,v is the number of walks of length k from u to v .
Pn p 1/p Let kxkp , ( i=1 |xi| ) . Then for any 1 ≤ p ≤ q < ∞, it holds that
1/p−1/q kxkq ≤ kxkp ≤ n · kxkq.
Lemma 4. For any graph G with m edges, the number of cycles of length k in G is bounded by O mk/2.
Proof. Let A be the adjacency matrix of G with eigenvalues λ1, . . . , λn. Thus the number k Pn k of Ck (cycles of length k) in G is bounded by tr A /(2k) = i=1 λi /(2k). For any k ≥ 3, it holds that
n !1/k n !1/k n !1/2 X k X k X 2 1/2 λi ≤ |λi| ≤ |λi| = (2 · m) . i=1 i=1 i=1