On graphs with large third eigenvalue

TL;DR

The paper investigates the maximum possible value of the third eigenvalue $\lambda_3$ of a graph's adjacency matrix, focusing on the conjectured bound $\lambda_3(G) \le |V(G)|/3$ and providing constructions like $H_{a,b}$ that achieve $\lambda_3 = |V|/3 - 1$. It proves the bound for several graph families (strongly regular, regular line graphs, and abelian Cayley and semi-Cayley graphs, among others) and extends the question to weighted graphs, establishing near-equivalence to the unweighted case via Weyl-type inequalities and a finite reduction to an $11\times11$ template. The work also analyzes extremisers of $\lambda_{n-1}$ in weighted settings, deriving structural results and a geometric framework, and discusses limits via graphons and the $H_{a,b}$ family as a bridge between equality cases and limiting behavior. Overall, the results advance understanding of when $\lambda_3$ reaches the conjectured threshold and illuminate the structure of extremal graphs through algebraic, combinatorial, and analytic methods.

Abstract

Given a graph $G$, let $λ_3$ denote the third largest eigenvalue of its adjacency matrix. In this paper, we prove various results towards the conjecture that $λ_3(G) \le \frac{|V(G)|}{3}$, motivated by a question of Nikiforov. We generalise the known constructions that yield $λ_3(G) = \frac{|V(G)|}{3} - 1$ and prove the inequality holds for $G$ strongly regular, a regular line graph or a Cayley graph on an abelian group. We also consider the extended problem of minimising $λ_{n-1}$ on weighted graphs and reduce the existence of a minimiser with simple final eigenvalue to a vertex multiplication of a graph on 11 vertices. We prove that the minimal $λ_{n-1}$ over weighted graphs is at most $O(\sqrt{n})$ from the minimal $λ_{n-1}$ over unweighted graphs.

Figures (4)

• Figure 1: $H_{4,0}$, $H_{3,1}$ and $H_{2,2}$ respectively.
• Figure 2: Two parallelograms.
• Figure 3: Examples of Pivalous graphs. From left to right: $Pi_3$, $Pi_6$, $Pi_{10}$, $Pi_{20}$, $Pi_{30}$.
• Figure 4: Visually demonstrating convergence with $Pi_3, Pi_6, Pi_{10}, Pi_{50}, Pi_{1000}$.

Theorems & Definitions (62)

• Theorem 1.1: Nikiforov NikiforovEvals
• Theorem 1.2: Nikiforov NikiforovEvals
• Theorem 1.3: Linz LinzImproved
• Conjecture 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Theorem 2.2
• proof