Equiangular lines and spherical codes in Euclidean space

Balla, Igor; Dräxler, Felix; Keevash, Peter; Sudakov, Benny

doi:10.1007/s00222-017-0746-0

Cited by 51 publications

(99 citation statements)

References 21 publications

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“…One can also consider the related problem of, for fixed α ∈ (0, 1), finding N α (d), the maximum number of lines in R d through the origin with pairwise angle arccos α. Motivated by a conjecture of Bukh [4], the asymptotic behaviour of N α (d) was recently shown to be linear in d [3,20].In Table 1 below, we give the currently known (including the improvements from this paper) values or lower and upper bounds for N (d) for d at most 23.…”

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confidence: 99%

Equiangular lines in Euclidean spaces

Greaves

Koolen

Munemasa

et al. 2016

Journal of Combinatorial Theory, Series A

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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.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table

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confidence: 99%

Equiangular lines in Euclidean spaces

Greaves

Koolen

Munemasa

et al. 2016

Journal of Combinatorial Theory, Series A

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“…Seidel matrices were introduced by Van Lint and Seidel [62] and studied by many people due to their interesting properties and connections to equiangular lines, two-graphs, strongly regular graphs, mutually unbiased bases and so on (see [10,Section 10.6] and [4,45,73] for example). The connection between Seidel matrices and equiangular lines is perhaps best summarized in [10, p.161]:…”

Section: Seidel Matricesmentioning

confidence: 99%

“…[60,Theorem 3.4]) proved that if N α (d) ≥ 2d, then 1/α is an odd integer. Bukh [12] proved that N α (d) ≤ c α d, where c α is a constant depending only on α. Balla, Dräxler, Keevash and Sudakov [4] improved this bound and showed that for d sufficiently large and α = 1/3, When 1/α is not a totally real algebraic integer, then N α (d) = d. Jiang and Polyanskii [56] studied the set T = {α | α ∈ (0, 1), lim sup d→∞ N α (d)/d > 1} and showed that the closure of T contains the closed interval [0, 1/ √ 5 + 2] using results of Shearer [75] on the spectral radius of unsigned graphs.…”

Section: Seidel Matricesmentioning

confidence: 99%

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Open problems in the spectral theory of signed graphs

Belardo

Cioabă

Koolen

et al. 2019

Art Discrete Appl. Math.

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Signed graphs are graphs whose edges get a sign +1 or −1 (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs.Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.

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“…A Seidel switching with respect to U transforms G to a graph H by deleting the edges between U and W and adding an edge between vertices u ∈ U and w ∈ W if (u, w) / ∈ E(G). For more details on Seidel matrices and related topics, see [15,11,12,1] and the references there. Seidel switching is an equivalence relation and we say that G and H are switching equivalent.…”

Section: Introductionmentioning

confidence: 99%

Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum

Berman

Shaked-Monderer

Singh

et al. 2019

Linear Algebra and its Applications

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It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph G on more than one vertex does not determine the graph, since any graph obtained from G by Seidel switching has the same Seidel spectrum. We consider G to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to G. It is shown that any graph which has the same spectrum as a complete k-partite graph is switching equivalent to a complete k-partite graph, and if the different partition sets sizes are p 1 , . . . , p l , and there are at least three partition sets of each size p i , i = 1, . . . , l, then G is determined, up to switching, by its Seidel spectrum. Sufficient conditions for a complete tripartite graph to be determined by its Seidel spectrum are discussed, and a conjecture is made on complete tripartite graphs on more than 18 vertices.

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Equiangular lines and spherical codes in Euclidean space

Cited by 51 publications

References 21 publications

Equiangular lines in Euclidean spaces

Equiangular lines in Euclidean spaces

Open problems in the spectral theory of signed graphs

Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum

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