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 $$N_{\alpha ,\beta }(d)$$...
| Main Authors: | , , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2023
|
| Online Access: | https://hdl.handle.net/1721.1/151162 |
| _version_ | 1826201039683977216 |
|---|---|
| author | Jiang, Zilin Tidor, Jonathan Yao, Yuan Zhang, Shengtong Zhao, Yufei |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Jiang, Zilin Tidor, Jonathan Yao, Yuan Zhang, Shengtong Zhao, Yufei |
| author_sort | Jiang, Zilin |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T11:45:42Z |
| format | Article |
| id | mit-1721.1/151162 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:45:42Z |
| publishDate | 2023 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1511622024-01-12T18:34:13Z Spherical Two-Distance Sets and Eigenvalues of Signed Graphs Jiang, Zilin Tidor, Jonathan Yao, Yuan Zhang, Shengtong Zhao, Yufei Massachusetts Institute of Technology. Department of Mathematics 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. 2023-07-25T18:46:06Z 2023-07-25T18:46:06Z 2023-07-21 2023-07-23T03:10:50Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/151162 Jiang, Zilin, Tidor, Jonathan, Yao, Yuan, Zhang, Shengtong and Zhao, Yufei. 2023. "Spherical Two-Distance Sets and Eigenvalues of Signed Graphs." PUBLISHER_CC en https://doi.org/10.1007/s00493-023-00002-1 Creative Commons Attribution http://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Jiang, Zilin Tidor, Jonathan Yao, Yuan Zhang, Shengtong Zhao, Yufei Spherical Two-Distance Sets and Eigenvalues of Signed Graphs |
| title | Spherical Two-Distance Sets and Eigenvalues of Signed Graphs |
| title_full | Spherical Two-Distance Sets and Eigenvalues of Signed Graphs |
| title_fullStr | Spherical Two-Distance Sets and Eigenvalues of Signed Graphs |
| title_full_unstemmed | Spherical Two-Distance Sets and Eigenvalues of Signed Graphs |
| title_short | Spherical Two-Distance Sets and Eigenvalues of Signed Graphs |
| title_sort | spherical two distance sets and eigenvalues of signed graphs |
| url | https://hdl.handle.net/1721.1/151162 |
| work_keys_str_mv | AT jiangzilin sphericaltwodistancesetsandeigenvaluesofsignedgraphs AT tidorjonathan sphericaltwodistancesetsandeigenvaluesofsignedgraphs AT yaoyuan sphericaltwodistancesetsandeigenvaluesofsignedgraphs AT zhangshengtong sphericaltwodistancesetsandeigenvaluesofsignedgraphs AT zhaoyufei sphericaltwodistancesetsandeigenvaluesofsignedgraphs |