Equiangular lines in Euclidean spaces

Gary Greaves; Jacobus H. Koolen; Akihiro Munemasa; Ferenc Szöllosi

doi:10.1016/j.jcta.2015.09.008

Equiangular lines in Euclidean spaces

Gary Greaves, Jacobus H. Koolen, Akihiro Munemasa, Ferenc Szöllosi

INF - Computer and Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

48 Citations (Scopus)

Abstract

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d= 14 and d= 16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.

Original language	English
Pages (from-to)	208-235
Number of pages	28
Journal	Journal of Combinatorial Theory. Series A
Volume	138
DOIs	https://doi.org/10.1016/j.jcta.2015.09.008
Publication status	Published - 2016 Feb 1

Keywords

Equiangular lines
Seidel matrix
Switching
Two-graph

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Access to Document

10.1016/j.jcta.2015.09.008

Cite this

@article{7216d6d7061c4369b28a5c22bf22c6af,

title = "Equiangular lines in Euclidean spaces",

abstract = "We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d= 14 and d= 16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.",

keywords = "Equiangular lines, Seidel matrix, Switching, Two-graph",

author = "Gary Greaves and Koolen, {Jacobus H.} and Akihiro Munemasa and Ferenc Sz{\"o}llosi",

note = "Funding Information: We are greatly indebted to Andries Brouwer and Donald Taylor for their helpful remarks regarding questions raised during the preparation of this manuscript. We are grateful for Alexander Barg and Wei-Hsuan Yu for sharing their notes prior to publication. We also thank the reviewers for their thorough work. This work was supported in part by the Hungarian National Research Fund OTKA K-77748 ; by the JSPS KAKENHI Grant Numbers 24 ⋅ 02789 , and 24 ⋅ 02807 ; and by the “100 Talents Program” of the Chinese Academy of Sciences . Part of the computational results in this research were obtained using supercomputing resources at Cyberscience Center, Tohoku University. Publisher Copyright: {\textcopyright} 2015 Elsevier Inc.",

year = "2016",

month = feb,

day = "1",

doi = "10.1016/j.jcta.2015.09.008",

language = "English",

volume = "138",

pages = "208--235",

journal = "Journal of Combinatorial Theory - Series A",

issn = "0097-3165",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Equiangular lines in Euclidean spaces

AU - Greaves, Gary

AU - Koolen, Jacobus H.

AU - Munemasa, Akihiro

AU - Szöllosi, Ferenc

N1 - Funding Information: We are greatly indebted to Andries Brouwer and Donald Taylor for their helpful remarks regarding questions raised during the preparation of this manuscript. We are grateful for Alexander Barg and Wei-Hsuan Yu for sharing their notes prior to publication. We also thank the reviewers for their thorough work. This work was supported in part by the Hungarian National Research Fund OTKA K-77748 ; by the JSPS KAKENHI Grant Numbers 24 ⋅ 02789 , and 24 ⋅ 02807 ; and by the “100 Talents Program” of the Chinese Academy of Sciences . Part of the computational results in this research were obtained using supercomputing resources at Cyberscience Center, Tohoku University. Publisher Copyright: © 2015 Elsevier Inc.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d= 14 and d= 16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.

AB - We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d= 14 and d= 16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.

KW - Equiangular lines

KW - Seidel matrix

KW - Switching

KW - Two-graph

UR - http://www.scopus.com/inward/record.url?scp=84946882424&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946882424&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2015.09.008

DO - 10.1016/j.jcta.2015.09.008

M3 - Article

AN - SCOPUS:84946882424

VL - 138

SP - 208

EP - 235

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

ER -

Equiangular lines in Euclidean spaces

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this