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 \ ₁²+₂² (1-1 (G) ) 2m\ where ₁ and ₂ denote the two largest eigenvalues of the adjacency matrix A (G), and 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-) triangles for some > 0. In particular, we show that the conjecture is true for planar graphs, book-free graphs and cycle-free graphs.