Equiangular lines and large multiplicity of fixed second eigenvalue

Carl Schildkraut

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arXiv · 2302.12230

Equiangular lines and large multiplicity of fixed second eigenvalue

Carl Schildkraut

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Abstract

Answering a question of Jiang and Polyanskii as well as Jiang, Tidor, Yao, Zhang, and Zhao, we show the existence of infinitely many angles $\theta$ for which the maximum number of lines in $\mathbb R^n$ meeting at the origin with pairwise angles $\theta$ exceeds $n+\Omega(\log\log n)$ but is at most $n+o(n)$. To accomplish this, we construct, for various real $\lambda$ and integer $d$, $d$-regular graphs with second eigenvalue exactly $\lambda$ and arbitrarily large second eigenvalue multiplicity. Central to our construction is a distribution on factors of bipartite graphs which possesses concentration properties.

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Carl Schildkraut. 2023-02-23. Equiangular lines and large multiplicity of fixed second eigenvalue. https://arxiv.org/abs/2302.12230

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