Bollob\'{a}s-Nikiforov conjecture holds asymptotically almost surely

TL;DR

This paper proves that the Bollobás-Nikiforov conjecture about eigenvalues and clique number holds asymptotically almost surely for large random graphs, confirming the conjecture in a probabilistic setting.

Contribution

The paper demonstrates that the conjecture is true with high probability for large random graphs, establishing an asymptotic almost sure result.

Findings

Conjecture holds asymptotically almost surely for large random graphs.

Eigenvalue inequality is confirmed in probabilistic models.

Supports the conjecture's validity in the limit for random graph sequences.

Abstract

Bollob\'{a}s 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 $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq 2e(G)\left(1-\frac{1}{\omega(G)}\right), $ where $\lambda_{1}$ and $\lambda_{2}$ 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.