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.