Exercise

							      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