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

Zilin Jiang, Alexandr Polyanskii

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4 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)393-421
Number of pages29
JournalIsrael Journal of Mathematics
Volume236
Issue number1
DOIs
StatePublished - Mar 1 2020
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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