# Homework #1

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 is the same as and is defined as the conjuniction of and .]

### Problems

1. Let be the adjacency matrix of a -regular graph . Prove that is bipartite if and only if has as an eigenvalue.
2. 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.

3. Let for some integer . Prove that if is the complete binary tree of height , we have

where again refers to the combinatorial Laplacian.

4. 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.

5. 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.