Graphs with high second eigenvalue multiplicity

TL;DR

This paper explores the maximum possible second eigenvalue multiplicity in bounded degree graphs, providing new lower bounds and constructions that challenge existing upper bounds, revealing fundamental limitations of current techniques.

Contribution

The authors construct graphs with higher second eigenvalue multiplicity than previously known, establishing new lower bounds and demonstrating the tightness of existing upper bounds up to a constant factor.

Findings

Constructed graphs with second eigenvalue multiplicity of order rac{rac{n}{\u007log n}}

Cayley graphs with second eigenvalue multiplicity at least n^{2/5}-1

Upper bounds on approximate second eigenvalue multiplicity are tight up to a constant factor

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.