# Support of Closed Walks and Second Eigenvalue Multiplicity of Regular Graphs

@article{McKenzie2020SupportOC, title={Support of Closed Walks and Second Eigenvalue Multiplicity of Regular Graphs}, author={Theo McKenzie and Peter Michael Reichstein Rasmussen and Nikhil Srivastava}, journal={ArXiv}, year={2020}, volume={abs/2007.12819} }

We show that the multiplicity of the second adjacency matrix eigenvalue of any connected $d-$regular graph is bounded by $O(n d^{7/5}/\log^{1/5-o(1)}n)$ for any $d$, and by $O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ when $d\ge \log^{1/4}n$ and the graph is simple. In fact, the same bounds hold for the number of eigenvalues in any interval of width $\lambda_2/\log_d^{1-o(1)}n$ containing the second eigenvalue $\lambda_2$. The main ingredient in the proof is a polynomial (in $k$) lower bound on the…

## One Citation

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