Spherical Two-Distance Sets and Eigenvalues of Signed Graphs
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493_2023_Article_2.pdf
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Author(s)
Jiang, Zilin
Tidor, Jonathan
Yao, Yuan
Zhang, Shengtong
Zhao, Yufei
Date Issued
July 21, 2023
Publisher
Springer Berlin Heidelberg
Citation
Jiang, Zilin, Tidor, Jonathan, Yao, Yuan, Zhang, Shengtong and Zhao, Yufei. 2023. "Spherical Two-Distance Sets and Eigenvalues of Signed Graphs."
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00493-023-00002-1
