This homework is due in two weeks on Wednesday, April 18th. You make work in small groups of 2 or 3. You should reference any discussions with others, books, lecture notes, or other resources that you use in coming up with a solution.

[**Reminder:** The notation

### Problems

- Let
be the adjacency matrix of a -regular graph . Prove that is bipartite if and only if has as an eigenvalue. - Consider the
two-dimensional grid graph . Let . Prove that where

refers to the second eigenvalue of the combinatorial Laplacian. (Prove this without using the Spielman-Teng theorem for planar graphs.) For the lower bound, it may help to remember that , where is the path on vertices. - Let
for some integer . Prove that if is the complete binary tree of height , we have where again

refers to the combinatorial Laplacian. - Let
be an arbitrary graph. Recall that Let

be the second eigenvalue of the combinatorial Laplacian on . Prove that,

and

where both minimums are over non-constant functions.

- Let
be a -regular graph and set , where is the adjacency matrix of . Let be defined as in the notes for lecture 3. Let be the eigenvalues of . Prove that by evaluating

in two different ways.