Exercises set #1

1. 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.

2. Exercises set #1 (cont.) Eigenvalues under matrix operations 2: Let A, B be a symmetric matrices. Prove: • i(At) = i(A)tfor 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.

3. 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

4. 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 #2 (cont.)

5. 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, greplacing (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. Exercises set #3