Exercises set #1
The notation refers to the lecture slides.
Eigenvalues and basic graph properties.
1: Assume G is a d-regular graph on n nodes.
AG is its normalized adj matrix. Prove that
- G is connected iff 1 > 2 . In fact,
# connected components = # 1 e-values
- a connected G is bipartite iff n = -1
- diameter(G) < 2log n / log(1/2)
*Try
removing the factor of 2 in this upper bound.
Exercises set #1 (cont.)
Eigenvalues under matrix operations
2: Let A, B be a symmetric matrices. Prove:
- i(At) = i(A)t for every integer t.
Use this to show that for a d-regular graph G,
tr(AG) = i i(AG)2 = n/d. Try using it to lower
bound (G) for any d-regular graph.
- i,j(AB) = i(A) j(B) , where  is the tensor
product of the two matrices.
Exercises set #2
Cayley graphs
H finite group, SH, symmetric. The Cayley graph
Cay(H;S) has xsx for all xH, sS.
Cay(Cn ; {-1,1})
Cay(F2n ; {e1,e2,…,en})
1. Compute all eigenvalues of these graphs
Hint: they are diagonalized respectively by the
Fourier Transform and Hadamard-Walsh matrices:
FT(i,j)=wij wn=1
HW(S,T)=(-1)ST S,T[n]
Exercises set #2 (cont.)
Abelian Cayley graphs
2. Prove that if H is Abelian, then C(H;S) cannot
be an expander for any |S|=d=O(1), |H| 
Hint: Give two proofs:
- For general H, use the combinatorial definition.
Use commutativity to show that the number of
different elements generated by length t words in S
grows too slowly as a function of t.
-*For H=Cn, use the spectral definition.
Prove that if S={a1,a2,..,ad}, then the eigenvalues are
i = ia_1+ia_2+…+ia_d and then by a pigeonhole
argument that for some i0, i must be close to 1.
Exercises set #3
Poincare Inequality
1. Prove the Poincare inequality: for every regular
graph G, and every real function f on its vertices
(1-c(G)) Eu,v [(f(u)-f(v))2]  Eu~v [(f(u)-f(v))2]
with c(G)=2(G) (the left expectation is over all pairs
u,v, and the right over all neighboring pairs u,v)
2. Extend the theorem to functions f whose range
is Rn, replacing (f(u)-f(v))2 with ||f(u)-f(v)||2
(hint: the inequality decomposes to n 1-dim. ones)
3. Extend the theorem to pairs of functions f, g
replacing (f(u)-f(v))2 with (f(u)-g(v))2, with c(G)=(G)
4. Prove that these choices of c(G) are tight.