Spherical two-distance sets and eigenvalues of signed graphs

Abstract

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let N α,β (d) denote the maximum number of unit vectors in R d where all pairwise inner products lie in {α, β}. For fixed −1 ≤ β < 0 ≤ α < 1, we propose a conjecture for the limit of N α,β (d)/d as d → ∞ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when α + 2β < 0 or. Our work builds on our recent resolution of the problem in the case of α = −β (corresponding to equiangular lines). It is the first determination of lim d→∞ N α,β (d)/d for any nontrivial fixed values of α and β outside of the equiangular lines setting.

Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below

“…Conjecture A was confirmed in [17] for p ≤ 2 or λ ∈ 1, √ 2, √ 3 . Building on the framework developed there, we establish Conjecture A for all λ < λ * as an application of Corollary 7.…”

“…For the general case −1 ≤ β < 0 ≤ α < 1, in their subsequent work [17], Jiang et al proposed a conjecture on the limit of N α,β (d)/d as d → ∞. To state their conjecture, we need the following spectral graph quantity.…”

“…Conjecture A asserts that the constructions in Proposition 44 are optimal up to an error sublinear in d. A framework to bound N α,β (d) from above was formulated in [17].…”

Graphs with high second eigenvalue multiplicity

“…We construct a new graph G from H. Let the vertex set V be {(i, v) : v ∈ V (H), i ∈ {1, 2, 3, 4}}. Let the graph H 1 be the disjoint union of n copies of K 4 with vertices {(i, v) : i ∈ {1, 2, 3, 4}} for each v ∈ V (H), and let the graph H 2 be the disjoint union of 4 copies of H with vertices {(i, v) v ∈ V (H)} for each i ∈ [4]. We construct G = G(H, ℓ) as graph obtained by first overlaying H 1 and H 2 and then replacing each edge of H 2 with a path of length ℓ.…”

“…(d) As noted in [3], Theorem 1.1 also applies to the j-th eigenvalue of G for any fixed j. While the second eigenvalue multiplicity result is sufficient for equiangular lines, the j-th eigenvalue multiplicity result is needed for a generalization to spherical two-distance sets [4]. From a spectral graph theory perspective, the second eigenvalue multiplicity result is the most interesting case and is our focus here.…”