# Lecture 6

Here are the slides, and here’s a proof of the crossing number inequality.

From this paper, we know that for an n-vertex planar graph with maximum degree ${\Delta}$, we have for every k, the eigenvalue bound

${\lambda_k \leq O(\Delta k/n)}$

Open question: What about the eigenvalue gaps? I conjecture that the stronger bound ${\lambda_{k+1} - \lambda_k \leq O(\Delta/n)}$ holds for every k. I believe it is open even for k=2.

## One thought on “Lecture 6”

1. Cyrus Rashtchian says:

I’ve looked around a little for results on eigenvalue gaps, and it seems like the primary focus in most fields is on the largest/smallest nontrivial eigenvalues/eigenfunctions. However, I imagine other people have been interested in eigenvalue gaps and the distribution of eigenvalues for certain operators. Do you know of any pointers to research in these areas?