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

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.