A note on the Bollobás-Nikiforov conjecture

Abstract

Bollobás and Nikiforov [2] proposed a conjecture that for any non-complete graph G with m edges and clique number ω, the following inequality holds:${\lambda }_1^2+{\lambda }_2^2\le 2(1-\frac{1}{\omega })m,$ where ${\lambda }_1$ and ${\lambda }_2$ are the two largest eigenvalues of the adjacency matrix $A\left(G\right)$. Later, Elphick, Linz, and Wocjan [6] proposed a generalization of this conjecture. In this paper, we prove that the conjecture proposed by Bollobás and Nikiforov holds for both line graphs and graphs with at most $\frac{\sqrt{6}}{27}{m}^{\frac{3}{2}}$ triangles, and that the generalized conjecture holds for both line graphs with additional conditions and graphs with not many triangles, which extends and strengthens some known results.