# Exercise

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```							      Expander                 Graphs                  and                  Their           Applications

Exercise 4
Lecturer: Amir Shpilka                                                Hand in date: March 11th, 2007

1. Analysis of the Zigzag product.
In this question you will prove that the zigzag product of two expander graphs is an
¯                                  ¯
expander graph. Let G be an [n, D, λG ] expander graph, and H a [D, d, λH ] expander
graph (λ¯ G is the normalized second eigenvalue of G in absolute value). Prove that
¯    ¯
G ×z H (the zigzag product of G and H) is an [nD, d2 , λG + λH ] graph.
Instruction:
It will be helpful to us to think of vectors in RnD as of n × D matrices. That is,
each vector u ∈ RnD is denoted with u = (ui,j )i∈[n],j∈[d] . Consider the following two
mappings on vectors in RnD = Rn ⊗ RD :
For u ∈ RnD and i ∈ [n] let u(i) ∈ RD be the vector (u(i) )j = ui,j (that is, u(i) is the
i-th row of u).
Let σ : RnD → Rn be deﬁned as (σu)i = ΣD ui,j (we multiply the matrix u with the
j=1
all 1 vector of length D).
n                                         (i)          (i)
Note that u =      i=1 ei   ⊗ u(i) . Denote with u⊥ and u                       the ”parts” of u that are
(i)   (i)
perpendicular to 1 (the all 1 vector) and parallel to it. In particular u(i) = u⊥ + u .
n             (i)                   n              (i)
Denote u =      i=1 ei   ⊗u         and u⊥ =        i=1 ei   ⊗ u⊥ . Prove that

1
u =     (σu) ⊗ 1.
D

Let Az be the normalized adjacency matrix of the zigzag product (that is, the adja-
cency matrix divided by the degree). Show that Az can be written as

Az = (I ⊗ AH ) · R · (I ⊗ AH ),

where AH is the normalized adjacency matrix of H, and R is a permutation matrix.
For simplicity we denote B = I ⊗ AH . Notice that B is a symmetric matrix. In order
to estimate the second eigenvalue of Az we need to bound the norm of Az · u for every
1 ⊥ u ∈ RnD .
From now on we assume that 1 ⊥ u ∈ RnD , and that u = 1. Prove that

(a) σu ⊥ 1.
(b) Bu = u .
¯
(c) Bu⊥ ≤ λH · u⊥ .
1                           ¯
λG             2      ¯           2.
(d) | Ru , u | =   D|    AG · σu, σu | ≤       D     · σu          = λG · u

4-1
Conclude that
¯
| Az u, u | ≤ λG · u      2      ¯
+ 2λ H · u ⊥ · u         ¯
+ λ2 · u ⊥    2     ¯        ¯ ¯
≤ λH + max(λG , λ2 ).
H                              H

¯           ¯
Bonus: Show that the second eigenvalue is upper bounded by 1 − 1 (1 − λG ) · (1 − λ2 ).
2                    H

2. Non-regular expanders
Let G be a graph on n vertices. Let di be the degree of the i-th vertex and A be its
adjacency matrix. Deﬁne T to be a diagonal matrix with Ti,i = di . Let L = T − A
and set
L = T −1/2 · L · T −1/2
−1
with the convention that Ti,i = 0 when di = 0 (i.e. when there is an isolated vertex).
In particular                       
 1
        i=j
Li,j =   √−1    i∼j
 di dj
0      otherwise


For a set S ⊂ [n] deﬁne vol(S) =          i∈S   di . Let the (normalized) Cheeger constant be
deﬁned as
E(S, S c )
hG = min                          .
S⊂[n] min(vol(S), vol(S c ))

From now on we assume that G is connected.

¯            ¯
(a) Prove that L is positive semi deﬁnite, and has eigenvalues 0 ≤ λ2 ≤ . . . ≤ λn .
(b) Show that T 1/2 1 is the eigenvector with eigenvalue of 0, and conclude that

¯                   Lv, v
λ2 =      inf             .
v⊥T 1/2 1   v, v

¯
(c) Prove that 2hG ≥ λ2 .
(d) For this item we assume that G is not a bipartite graph. Consider the random
walk where we move to a random neighbor with the uniform probability. Prove
1
that the stationary distribution for this walk is given by vol(G) T 1. Prove that no
matter from which initial distribution we begin our walk with we have that after
k steps the L2 -distance from the stationary distribution is bounded from above
√
¯                   ¯         ¯       ¯
by (1 − λ)k maxi√di , where λ = min(λ2 , 2 − λn ).
minj   dj

(e) Prove that for every two subsets X, Y ⊂ [n] we have that

vol(X) · vol(Y )        ¯
|E(X, Y )| −                       ≤ (1 − λ) ·           vol(x) · vol(Y ).
vol(G)

(f) Verify to yourself (i.e. no need to submit a solution) that hG <                   ¯
2λ2 .

4-2

```
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