Spherical Two-Distance Sets and Eigenvalues of Signed Graphs

Abstract

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

$$N_{\alpha ,\beta }(d)$$

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.