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On the first two eigenvalues of regular graphs

Shengtong Zhang

Abstract

Let $G$ be a regular graph with $m$ edges, and let $μ_1, μ_2$ denote the two largest eigenvalues of $A_G$, the adjacency matrix of $G$. We show that, if $G$ is not complete, then $$μ_1^2 + μ_2^2 \leq \frac{2(ω- 1)}ω m$$ where $ω$ is the clique number of $G$. This confirms a conjecture of Bollobás and Nikiforov for regular graphs. We also show that equality holds if and only if $G$ is either a balanced Turán graph or the disjoint union of two balanced Turán graphs of the same size.

On the first two eigenvalues of regular graphs

Abstract

Let be a regular graph with edges, and let denote the two largest eigenvalues of , the adjacency matrix of . We show that, if is not complete, then where is the clique number of . This confirms a conjecture of Bollobás and Nikiforov for regular graphs. We also show that equality holds if and only if is either a balanced Turán graph or the disjoint union of two balanced Turán graphs of the same size.
Paper Structure (5 sections, 3 theorems, 41 equations)

This paper contains 5 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Recap of Previous Work
  3. Proof of Conjecture 1.1 for regular graphs
  4. The case of equality
  5. Difficulties Ahead

Key Result

Theorem 1.2

Suppose $G$ is a regular graph with clique number $\omega$, and is not the complete graph $K_{\omega}$. Then The equality holds if and only if 1) $G$ is a Turán graph $T(n, \omega)$ with $n$ divisible by $\omega$ or 2) $G$ is the disjoint union $T(n / 2, \omega) \sqcup T(n / 2, \omega)$ of two Turán graphs with $n$ divisible by $2\omega$.

Theorems & Definitions (5)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 2.2
  • proof