Spherical Two-Distance Sets and Eigenvalues of Signed Graphs

Abstract

AbstractWe 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 $$N_{\alpha ,\beta }(d)$$ N α , β ( d ) denote the maximum number of unit vectors in $${\mathbb {R}}^d$$ R d where all pairwise inner products lie in $$\{\alpha ,\beta \}$$ { α , β } . For fixed $$-1\le \beta<0\le \alpha <1$$ - 1 ≤ β < 0 ≤ α < 1 , we propose a conjecture for the limit of $$N_{\alpha ,\beta }(d)/d$$ N α , β ( d ) / d as $$d \rightarrow \infty $$ d → ∞ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $$\alpha +2\beta <0$$ α + 2 β < 0 or $$(1-\alpha )/(\alpha -\beta ) \in \{1, \sqrt{2}, \sqrt{3}\}$$ ( 1 - α ) / ( α - β ) ∈ { 1 , 2 , 3 } .Our work builds on our recent resolution of the problem in the case of $$\alpha = -\beta $$ α = - β (corresponding to equiangular lines). It is the first determination of $$\lim _{d \rightarrow \infty } N_{\alpha ,\beta }(d)/d$$ lim d → ∞ N α , β ( d ) / d for any nontrivial fixed values of $$\alpha $$ α and $$\beta $$ β outside of the equiangular lines setting.