Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Quanyu Tang

Abstract

For a graph $G$ of order $n$, let $$ λ_1(G)\ge \cdots \ge λ_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ λ_3(G)\le \frac{n}{3}-1, $$ 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 $$ λ_{n-1}(A)+λ_n(A)\ge -\frac{2n}{3}. $$ This in particular confirms a conjecture of Leonida and Li.

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

Abstract

For a graph of order , let be the eigenvalues of its adjacency matrix. We prove that every graph on vertices satisfies thereby solving a problem of Nikiforov. The bound is best possible whenever . Our proof is derived from a more general matrix result: if is a real symmetric matrix of order with for all off-diagonal entries and for all , then This in particular confirms a conjecture of Leonida and Li.
Paper Structure (3 sections, 4 theorems, 50 equations)

This paper contains 3 sections, 4 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. A rank-two projection bound
  3. Deduction of the graph-theoretic theorem

Key Result

Theorem 1.3

Let $n\ge 2$, and let $A=(a_{ij})$ be a real symmetric matrix of order $n$ with eigenvalues Assume that Then In particular, $\mu_{n-1}\ge -\frac{n}{3}$.

Theorems & Definitions (11)

  • Conjecture 1.2: Leonida--Li LeonidaLi2026
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:weighted']}
  • Corollary 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • ...and 1 more