Spherical Two-Distance Sets and Eigenvalues of Signed Graphs

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