TY - JOUR

T1 - Forbidden Subgraphs for Graphs of Bounded Spectral Radius, with Applications to Equiangular Lines

AU - Jiang, Zilin

AU - Polyanskii, Alexandr

N1 - Funding Information:
Supported in part by ISF grant no. 409/16, by the Russian Foundation for Basic Research through grant no. 15-01-03530 A, and by the Leading Scientific Schools of Russia through grant no. NSh-6760.2018.1. Acknowledgements
Funding Information:
Supported in part by Israel Science Foundation (ISF) grant nos 1162/15, 936/16.
Publisher Copyright:
© 2020, The Hebrew University of Jerusalem.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let ℱ(λ) be the family of connected graphs of spectral radius ≤ λ. We show that ℱ(λ) can be defined by a finite set of forbidden subgraphs if and only if λ<λ*:=2+5≈2.058 and λ ∉ {α2, α3, …}, where αm=βm1/2+βm−1/2 and βm is the largest root of xm+1 = 1+ x + … + xm−1. The study of forbidden subgraphs characterization for ℱ(λ) is motivated by the problem of estimating the maximum cardinality of equiangular lines in the n-dimensional Euclidean space ℝn family of lines through the origin such that the angle between any pair of them is the same. Denote by Nα(n) the maximum number of equiangular lines in ℝn with angle arccos α. We establish the asymptotic formula Nα(n) = cαn + Oα(1) for every Nα(n) = cαn+ Oα(1). In particular, α≥11+2λ*. Besides we show that N1/3(n)=2n+O(1)andN1/5(n),N1/(1+22)(n)=32n+O(1)., which improves a recent result of Balla, Dräxler, Keevash and Sudakov.

AB - The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let ℱ(λ) be the family of connected graphs of spectral radius ≤ λ. We show that ℱ(λ) can be defined by a finite set of forbidden subgraphs if and only if λ<λ*:=2+5≈2.058 and λ ∉ {α2, α3, …}, where αm=βm1/2+βm−1/2 and βm is the largest root of xm+1 = 1+ x + … + xm−1. The study of forbidden subgraphs characterization for ℱ(λ) is motivated by the problem of estimating the maximum cardinality of equiangular lines in the n-dimensional Euclidean space ℝn family of lines through the origin such that the angle between any pair of them is the same. Denote by Nα(n) the maximum number of equiangular lines in ℝn with angle arccos α. We establish the asymptotic formula Nα(n) = cαn + Oα(1) for every Nα(n) = cαn+ Oα(1). In particular, α≥11+2λ*. Besides we show that N1/3(n)=2n+O(1)andN1/5(n),N1/(1+22)(n)=32n+O(1)., which improves a recent result of Balla, Dräxler, Keevash and Sudakov.

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U2 - 10.1007/s11856-020-1983-2

DO - 10.1007/s11856-020-1983-2

M3 - Article

AN - SCOPUS:85082854846

SN - 0021-2172

VL - 236

SP - 393

EP - 421

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -