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Equiangular lines and eigenvalue multiplicities

Yufei Zhao

Abstract

Expository article on the problem of determining the maximum number of equiangular lines with a fixed angle, and the associated problem of second eigenvalue multiplicity in graphs.

Equiangular lines and eigenvalue multiplicities

Abstract

Expository article on the problem of determining the maximum number of equiangular lines with a fixed angle, and the associated problem of second eigenvalue multiplicity in graphs.

Paper Structure

This paper contains 18 sections, 8 theorems, 19 equations, 1 figure, 1 table.

Table of Contents

  1. Equiangular Lines
  2. Connection to Spectral Graph Theory
  3. Proof Outline
  4. Switching to a Bounded Degree Graph
  5. Sublinear Second Eigenvalue Multiplicity
  6. Further Questions
  7. Acknowledgments

Key Result

Theorem 2

Every connected bounded degree graph has sublinear second eigenvalue multiplicity. More precisely, the second largest eigenvalue of the adjacency matrix of an $n$-vertex connected graph with maximum degree $\Delta$ has multiplicity $O_\Delta(n/\log\log n)$.

Figures (1)

  • Figure 1: A key idea in the proof of \ref{['thm:mult']} is that removing a net of vertices (white dots) significantly lowers local spectral radii ($\lambda_1$ of the dark disk).

Theorems & Definitions (12)

  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Theorem 6: Switching to bounded degree
  • Lemma 7: Finding a small net
  • proof
  • Lemma 8: Net removal reduces spectral radius
  • proof
  • Lemma 9: Local vs global spectra
  • ...and 2 more