Graphs with high second eigenvalue multiplicity

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 bestknown lower bound is on the order of 𝑛 1∕3 (for infinitely many 𝑛), coming from Cayley graphs on PSL(2, 𝑞).Here we give a construction showing a lower bound of √ 𝑛∕ log 2 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.

Equiangular lines and large multiplicity of fixed second eigenvalue

“…We obtain this result via constructing a family of bounded-degree regular graphs with second eigenvalue exactly λ = (1 − α)/(2α) for various λ. The second eigenvalue multiplicity we construct is much smaller than that in [HSZZ22]. However, to our knowledge, this is the first result giving unbounded second eigenvalue multiplicity when the eigenvalue is fixed.…”

“…McKenzie, Rasmussen, and Srivastava [MRS21] improved the upper bound to n/ log 1/5−o(1) n in the case of regular graphs, and also provided sublinear upper bounds in the unbounded degree case. On the other side, Haiman, Schildkraut, Zhang, and Zhao [HSZZ22] gave a lower bound (i.e. a construction of bounded-degree graphs with large second eigenvalue multiplicity) of n 1/2−o(1) .…”

Subdivision and Graph Eigenvalues

“…. Then H t is a disjoint union of the spiders (K In [3], the authors relaxed the notion of eigenvalue multiplicity to counting the number of eigenvalues in subsets of R; this gives us a formulation to talk about eigenvalue multiplicities with regards to subdivision. Given I ⊂ R, let m G (I) be the number of eigenvalues of G (counting with multiplicities) that lie in I.…”

“…They used it to solve a long-standing conjecture about equiangular lines. In a follow-up article [3], Haiman et al, came up with a construction which showed that the upper bound on m(λ 2 , G) would be sharp if the notion of multiplicity was relaxed to the 'approximate second eigenvalue multiplicity.' This construction uses Cartesian product of graphs and then a subdivision of a subset of edges to show desired spectral properties.…”

“…The behaviour of the spectrum studied in relation to subdivision generally falls into two broad categories: either a single edge is subdivided, or all of the edges are subdivided. However, recently, subdividing a subset of edges has found application in generating graphs with high second eigenvalue multiplicities [3]. In this article, our goal is to analyze the asymptotic behaviour of eigenvalues of a graph G when a subset S ⊆ E(G) is subdivided sufficiently many times.…”