Strengthened upper bound on the third eigenvalue of graphs

TL;DR

This paper resolves a longstanding conjecture-like claim by proving the strengthened upper bound for the third eigenvalue proportion in graphs, specifically establishing that no graph sequence can achieve $(\lambda_{n-1}+\lambda_n)/n$ approaching $-\frac{\sqrt{2}}{2}$. The authors shift focus from $\lambda_3$ directly to the sum $\lambda_{n-1}+\lambda_n$, introducing a novel graph operation $^*$ that constrains minimising graphs to have limited clique and chromatic numbers and to resemble invariant, structured forms. They then restrict analysis to $n/2$-regular graphs within this invariant framework, deriving a detailed, multi-stage inequality argument (involving front/middle/back decompositions and Type 1/Type 2 eigenvectors) to show the desired bound strictly improves beyond the conjectured limit. A key consequence is supporting the plausibility of $c_3=\tfrac{1}{3}$ and highlighting a deep connection between extremal spectral properties and invariant graph families such as $H_{a,b}$ and $\Pi_n$, with potential alternative proofs via graphon theory. The work thus advances spectral extremal graph theory by both tightening a precise bound and clarifying the structural landscape of near-worst-case graphs.

Abstract

Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $λ_1 \ge λ_2 \ge \dots \ge λ_n$. The problem of bounding $λ_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov, who demonstrated strong upper and lower bounds for arbitrary $k$. Nikiforov also claimed a strengthened upper bound for $k \ge 3$, namely that $\frac{λ_k}{n} < \frac{1}{2\sqrt{k-1}} - \varepsilon_k$ for some positive $\varepsilon_k$, but omitted the proof due to its length. In this paper, we give a proof of this bound for $k = 3$. We achieve this by instead looking at $λ_{n-1} + λ_n$ and introducing a new graph operation which provides structure to minimising graphs, including $ω\le 3$ and $χ\le 4$. Then we reduce the hypothetical worst case to a graph that is $n/2$-regular and invariant under said operation. By considering a series of inequalities on the restricted eigenvector components, we prove that a sequence of graphs with $\frac{λ_{n-1} + λ_n}{n}$ converging to $-\frac{\sqrt{2}}{2}$ cannot exist.

Figures (8)

• Figure 1: Invariant families. $\overline{H_{5,2}}$ on the left, $\overline{Pi_{21}}$ on the right.
• Figure 2: From left to right: $G_4$, $A(G_4)$, the resultant $\mathbb{R}^2$ vectors, $Q$.
• Figure 3: From left to right: $G_6$, $A(G_6)$, the resultant $\mathbb{R}^2$ vectors, $Q$.
• Figure 4: Visually demonstrating the inequalities for $G_4$ (left) and $G_6$ (right).
• Figure 5: Phase 1 visualised, with $T = 0.05$. Overall and close-up view.

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Definition 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4