Problem Set 2
MATH 778C, Spring 2009, Cooper
Expiration: Wednesday March 18
You are awarded up to 20 points for each problem, 5 points for submitting
solutions in L
A
T
E
X, and 5 points per solution that is used for the answer key.
All answers must be fully rigorous – do not assume anything that you are not
sure everyone else in the class knew prior to Day 1 of this class. However, you
may cite without proof any result proven in detail in class.
1. Let S⊂Z×
pbe such that S=−S, where pis prime. Define the “circulant
graph” GSby V(GS) = Zpand {x, y} ∈ E(GS) iff there exists s∈Sso
that x+s=y. Define the kth discrete Fourier coefficient of a function
f:Zp→C, for k∈Zp, by
ˆ
f(k) = 1
√p
p−1
X
j=0
e2πijk/p f(j).
Show that
λ(G) = √p
|S|max
k6=0 |ˆχS(k)|,
where χSis the characteristic function of S. Hint: First prove that
{fk}k∈Zpis an orthonormal basis of CZp, where fk(j) = exp(2πijk/p)/√p.
2. Let Aand Bbe symmetric stochastic matrices. Prove that λ(A+B)≤
λ(A) + λ(B).
3. Prove that, for every n-vertex d-regular graph, there is some subset Sof n/2
vertices so that |E(S, S)| ≤ dn/4 + O(1). Conclude that no (n, d, ρ)-edge
expander family exists if ρ > 1/2.
4. Let Gbe an (n, D, ρ)-edge expander and G0be a (D , d, ρ0)-edge expander,
for ρ, ρ0>0. Prove that GrG0is a (nD, 2d, ρ2ρ0/1000)-edge expander.
5. A (countably) infinite locally finite graph Gis said to be an f-expander,
for f:N→R+, if
min
S:n≤|S|<∞|E(S, S)|= Θ(f(n))
Give an example of a 1-expander, an n2/3-expander and an n-expander.
(Of course, you need to provide proofs that they work.)
