Bollobás-Nikiforov Conjecture for graphs with not so many triangles

Bollobás and Nikiforov conjectured that for any graph $G\mathrm{\ne }{K}_n$ with $m$ edges $${\lambda }_1^2+{\lambda }_2^2\le (1-\frac{1}{\omega (G)})2m$$ where ${\lambda }_1$ and ${\lambda }_2$ denote the two largest eigenvalues of the adjacency matrix $A(G)$, and $\omega$ denotes the clique number of $G$. This conjecture was recently verified for triangle-free graphs by Lin, Ning and Wu and for regular graphs by Zhang. Elphick, Wocjan and Linz proposed a generalization of this conjecture. In this note, we verify this generalized conjecture for the family of graphs on $m$ edges, which contain at most $O(m^{1.5-\epsilon })$ triangles for some $\epsilon >0$. In particular, we show that the conjecture is true for planar graphs, book-free graphs and cycle-free graphs.