Bollob\'as-Nikiforov Conjecture for graphs with not so many triangles

Bollob\'as and Nikiforov conjectured that for any graph G ≠ Kₙ with m edges \[ λ_1^2+λ_2^2≤ ( 1-1/ω(G))2m\] where λ₁ and λ₂ denote the two largest eigenvalues of the adjacency matrix $A(G)$, and ω 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(m1.5-ε) triangles for some ε > 0. In particular, we show that the conjecture is true for planar graphs, book-free graphs and cycle-free graphs.