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
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
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.
for some integer . Prove that if is the complete binary tree of height , we have
refers to the combinatorial Laplacian.
be an arbitrary graph. Recall that
be the second eigenvalue of the combinatorial Laplacian on .
where both minimums are over non-constant functions.
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
in two different ways.