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arXiv · 2109.13131

Graphs with high second eigenvalue multiplicity

Milan Haiman·Carl Schildkraut·Shengtong Zhang·Yufei Zhao
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Abstract

Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of equiangular lines with fixed angles. It led to the natural question: what is the maximum second eigenvalue multiplicity of a connected bounded degree $n$-vertex graph? The best known upper bound is $O(n/\log\log n)$. The previously known best known lower bound is on the order of $n^{1/3}$ (for infinitely many $n$), coming from Cayley graphs on $\text{PSL}(2,q)$. Here we give constructions showing a lower bound on the order of $\sqrt{n/\log n}$. We also construct Cayley graphs with second eigenvalue multiplicity at least $n^{2/5}-1$. Earlier techniques show that there are at most $O(n/\log\log n)$ eigenvalues (counting multiplicities) within $O(1/\log n)$ of the second eigenvalue. We give a construction showing this upper bound on approximate second eigenvalue multiplicity is tight up to a constant factor. This demonstrates a barrier to earlier techniques for upper bounding eigenvalue multiplicities.

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Milan Haiman, Carl Schildkraut, Shengtong Zhang, Yufei Zhao. 2021-09-27. Graphs with high second eigenvalue multiplicity. https://arxiv.org/abs/2109.13131

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