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

TL;DR

This paper establishes a new general upper bound on the $k$-th eigenvalue of a graph's adjacency matrix, extending previous results and connecting to equiangular lines.

Contribution

It generalizes an existing approach to provide a universal upper bound for all $k \\geq 3$ using Gegenbauer polynomials, improving understanding of eigenvalue limits.

Findings

The bound is tight for $k=2,3,4,8,24$.

The approach uses positivity of Gegenbauer polynomials.

Connections are made to equiangular line problems.

Abstract

We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] 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 $\lambda_k(G)$ to questions about equiangular lines.