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 best known 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 constructions showing a lower bound on the order of 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.

Equiangular lines via matrix projection

“…Note that for normalized adjacency matrices and therefore also for regular graphs, McKenzie, Rasmussen, and Srivastava [56] improve on the approach of [47] to improve the upper bound to O ∆ (n/ log c n) for some constant c. However, they give reason to suggest that their methods cannot improve the bound beyond O ∆ (n/ log n). Moreover, Haiman, Schildkraut, Zhang, and Zhao [40] give a construction of a graph with max degree ∆ = 4 and m(λ 2 ) ≥ n/ log n, and they also point out that for bounded degree (∆ = O(1)) graphs, √ n is a natural barrier for group representation based constructions such as theirs.…”