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.

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One thought on “Lecture 6

  1. 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?

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