The Lemmens-Seidel conjecture and forbidden subgraphs

Cao, Meng Yue; Koolen, Jack H.; Lin, Yen-chi Roger; Yu, Wei

doi:10.1016/j.jcta.2021.105538

Cited by 4 publications

(5 citation statements)

References 14 publications

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“…Note that the maximum cardinalities remain the same from d = 7 to d = 15, and become a linear function in d for d ≥ 15. There is a similar phenomenon when α = 1/5: Cao et al [11] proved that In general, the asymptotic behaviors of N α (d) are also determined.…”

Section: Introductionmentioning

confidence: 81%

“…On the other hand, for a = 5 Cao et al [11] proved that As indicated in Remark 2.4, the alternative semidefinite constraints Q d k (X) is simpler than the original one developed by Bachoc and Vallentin [3] when the concerned object X is a spherical s-distance set. The alternative constraints may be also useful when concerning objects other than sets of equiangular lines.…”

Section: Further Questionsmentioning

confidence: 99%

“…There are at least two maximum constructions in R 15 . Meanwhile, Cao et al [11] proved that N 1/5 (d) = 276 for 23 ≤ d ≤ 185. It is known that there are at least two maximum constructions in R 185 , so it remains unknown that whether the maximum constructions in R d for 65 ≤ d ≤ 184 are unique.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

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Four-point semidefinite bound for equiangular lines

Kao¹,

Yu²

2022

Preprint

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A set of lines in R d passing through the origin is called equiangular if any two lines in the set form the same angle. We proved an alternative version of the three-point semidefinite constraints developed by Bachoc and Vallentin, and the multi-point semidefinite constraints developed by Musin for spherical codes. The alternative semidefinite constraints are simpler when the concerned object is a spherical s-distance set. Using the alternative four-point semidefinite constraints, we found the four-point semidefinite bound for equiangular lines. This result improves the upper bounds for infinitely many dimensions d with prescribed angles. As a corollary of the bound, we proved the uniqueness of the maximum construction of equiangular lines in R d for 7 ≤ d ≤ 14 with inner product α = 1/3, and for 23 ≤ d ≤ 64 with α = 1/5.

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Section: Introductionmentioning

confidence: 81%

Section: Further Questionsmentioning

confidence: 99%

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Four-point semidefinite bound for equiangular lines

Kao¹,

Yu²

2022

Preprint

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“…Indeed, various versions of discrete nodal domain theory have been developed in different contexts. A very useful context should be the signed graphs, whose spectral theory has led to a number of breakthroughs in theoretical computer science and combinatorial geometry, including the solutions to the sensitivity conjecture [34] and the open problems on equiangular lines [11,36,37]. In addition, signed graphs have many other practical applications on modeling biological networks, social situations, ferromagnetism, and general signed networks [4,5,31].…”

Section: Introductionmentioning

confidence: 99%

Nodal domain theorems for $p$-Laplacians on signed graphs

Ge,

Liu,

Zhang

2023

J. Spectr. Theory

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We establish various nodal domain theorems for p-Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph p-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of p-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs. In the particular case of p D 1, this leads to several identities relating variational eigenvalues and multi-way Cheeger constants. Intriguingly, our approach also leads to new results on usual graphs, including a weak version of Sturm's oscillation theorem for graph 1-Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of p-Laplacians with p > 1 on connected bipartite graphs.

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Section: Introductionmentioning

confidence: 99%

Nodal domain theorems for $p$-Laplacians on signed graphs

Ge¹,

Liu²,

Zhang³

2022

Preprint

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We establish various nodal domain theorems for generalized p-Laplacians on signed graphs, which unify most of the results on nodal domains of graph p-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we also obtain a higher order Cheeger inequality that relates the variational eigenvalues of p-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs. Moreover, we show new results for 1-Laplacian on signed graphs, including a sharp upper bound of the number of nonzeros of certain eigenfunction corresponding to the k-th variational eigenvalues, and some identities relating variational eigenvalues and k-way Cheeger constants.

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The Lemmens-Seidel conjecture and forbidden subgraphs

Cited by 4 publications

References 14 publications

Four-point semidefinite bound for equiangular lines

Four-point semidefinite bound for equiangular lines

Nodal domain theorems for $p$-Laplacians on signed graphs

Nodal domain theorems for $p$-Laplacians on signed graphs

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