Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bollobás-Nikiforov conjecture holds asymptotically almost surely

Chunmeng Liu, Changjiang Bu

TL;DR

This work addresses the Bollobás-Nikiforov conjecture, which ties the sum of the squares of the two largest eigenvalues to the edge count and clique number of a graph. By leveraging known results on the largest and second-largest eigenvalues of random adjacency matrices, clique-number asymptotics in $G(n,p)$, and edge-count concentration via Hoeffding-type bounds, the authors prove that the conjecture holds asymptotically almost surely for random graphs. The main contribution is a rigorous probabilistic bound showing the inequality is satisfied with probability at least $1-\exp(-\epsilon^{2}p^{2}n(n-1))$ for large $n$. This result extends the conjecture's validity to the random-graph regime and reinforces its robustness in spectral graph theory contexts.

Abstract

Bollobás and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $ω(G)$, then $ λ_{1}^{2}+λ_{2}^{2}\leq 2e(G)\left(1-\frac{1}{ω(G)}\right), $ where $λ_{1}$ and $λ_{2}$ are the largest and the second largest eigenvalues of the adjacency matrix of $G$, respectively. In this paper, we prove that for a sequence of random graphs the conjecture holds true with probability tending to one as the number of vertices tends to infinity.

Bollobás-Nikiforov conjecture holds asymptotically almost surely

TL;DR

This work addresses the Bollobás-Nikiforov conjecture, which ties the sum of the squares of the two largest eigenvalues to the edge count and clique number of a graph. By leveraging known results on the largest and second-largest eigenvalues of random adjacency matrices, clique-number asymptotics in , and edge-count concentration via Hoeffding-type bounds, the authors prove that the conjecture holds asymptotically almost surely for random graphs. The main contribution is a rigorous probabilistic bound showing the inequality is satisfied with probability at least for large . This result extends the conjecture's validity to the random-graph regime and reinforces its robustness in spectral graph theory contexts.

Abstract

Bollobás and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph with edges and the clique number , then where and are the largest and the second largest eigenvalues of the adjacency matrix of , respectively. In this paper, we prove that for a sequence of random graphs the conjecture holds true with probability tending to one as the number of vertices tends to infinity.
Paper Structure (3 sections, 5 theorems, 38 equations)

This paper contains 3 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the theorem

Key Result

Theorem 1.1

Suppose that $G_{n}$ is a random graph as described above. For $0<\epsilon<1$, there exists an $n^{\prime}$ such that for all $n>n^{\prime}$, where $C=\epsilon^{2}p^{2}$.

Theorems & Definitions (8)

  • Conjecture 1
  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm_1']}