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 , we have for every k, the eigenvalue bound

**Open question:** What about the eigenvalue gaps? I conjecture that the stronger bound holds for every k. I believe it is open even for k=2.

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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?