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Upper bound on the $k$-th eigenvalue of a graph

Varun Sivashankar

Abstract

We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that addresses the case $k=3$ and generalize the upper bound for all $k \geq 3$ by using the positivity of Gegenbauer polynomials. The upper bound is tight for $k \in \{2,3,4,8,24\}$. We also highlight the close relation of $λ_k(G)$ to questions about equiangular lines.

Upper bound on the $k$-th eigenvalue of a graph

Abstract

We prove a general upper bound on the -th adjacency eigenvalue of a graph. For , we show that for every graph on vertices. We build on a recent approach that addresses the case and generalize the upper bound for all by using the positivity of Gegenbauer polynomials. The upper bound is tight for . We also highlight the close relation of to questions about equiangular lines.

Paper Structure

This paper contains 9 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Positive Kernels
  3. An $\ell_1$ bound for rank-$r$ orthogonal projections
  4. From projections to eigenvalues
  5. Concluding Remarks
  6. Tight Lower Bounds from Equiangular Lines
  7. Multiplicity of Second Eigenvalue
  8. Improving the Upper Bound
  9. Motivation for Choice of Polynomial

Key Result

Theorem 1.1

Let $k\ge 2$ be an integer. For every graph $G$ on $n$ vertices, Consequently, $c_k \leq \alpha_k$

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • Remark 1
  • Theorem 3.1
  • proof
  • Lemma 3
  • proof
  • ...and 8 more