A sharp upper bound on the third adjacency eigenvalue of a graph

TL;DR

This paper establishes a tight upper bound on the third largest eigenvalue of a graph's adjacency matrix, solving a problem posed by Nikiforov and confirming a related conjecture for symmetric matrices.

Contribution

It proves a sharp upper bound on the third adjacency eigenvalue of graphs, extending to a general matrix inequality that confirms a conjecture by Leonida and Li.

Findings

The bound is tight when the number of vertices is divisible by 3.

The eigenvalue inequality holds for all graphs with at least 3 vertices.

The result confirms a conjecture for symmetric matrices with bounded entries.

Abstract

For a graph $G$ of order $n$, let $$ \lambda_1(G)\ge \cdots \ge \lambda_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ \lambda_3(G)\le \frac{n}{3}-1, $$ thereby solving a problem of Nikiforov. The bound is best possible whenever $3\mid n$. Our proof is derived from a more general matrix result: if $A=(a_{ij})$ is a real symmetric matrix of order $n$ with $0\le a_{ij}\le 1$ for all off-diagonal entries and $a_{ii}\ge 0$ for all $i$, then $$ \lambda_{n-1}(A)+\lambda_n(A)\ge -\frac{2n}{3}. $$ This in particular confirms a conjecture of Leonida and Li.