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

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The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let \({\mathcal F}\left( \lambda \right)\) be the family of connected graphs of spectral radius ≤ λ. We show that \({\mathcal F}\left( \lambda \right)\) can be defined by a finite set of forbidden subgraphs if and only if \(\lambda > \lambda *: = \sqrt {2 + \sqrt 5 } \approx 2.058\) and λ ∉ {α2, α3, …}, where \({\alpha _m} = \beta _m^{1/2} + \beta _m^{ - 1/2}\) and βm is the largest root of xm+1 = 1+ x + … + xm−1. The study of forbidden subgraphs characterization for \({\mathcal F}\left( \lambda \right)\) 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 \({N_\alpha }\left( n \right) = {c_\alpha }n + {O_\alpha }\left( 1 \right)\). In particular, \(\alpha \ge {1 \over {1 + 2\lambda *}}\).

Besides we show that

$${N_{1/3}}\left( n \right) = 2n + O\left( 1 \right)\quad {\rm{and}}\quad {N_{1/5}}\left( n \right),\,{N_{1/(1 + 2\sqrt 2 )}}(n) = {3 \over 2}n\, + O\left( 1 \right).$$

, which improves a recent result of Balla, Dräxler, Keevash and Sudakov.

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Thanks to Boris Bukh for introducing equiangular lines to the first author, and to Jun Su and Sebastian Cioabă for useful correspondence. We wish to express our deep appreciation to the referee for meticulous reading, and for pointing out a mistake in Theorem 1 and many other inaccuracies in an earlier version of the manuscript. All remaining errors are ours.

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Correspondence to Zilin Jiang.

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The work was done when the authors were postdoctoral fellows at Technion—Israel Institute of Technology.

Supported in part by Israel Science Foundation (ISF) grant nos 1162/15, 936/16.

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.

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Jiang, Z., Polyanskii, A. Forbidden Subgraphs for Graphs of Bounded Spectral Radius, with Applications to Equiangular Lines. Isr. J. Math. 236, 393–421 (2020).

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