Bollobás-Nikiforov Conjecture for graphs with not so many triangles

Abstract

Bollobás and Nikiforov conjectured that for any graph $G\ne K_n$ with $m$ edges \[ λ_1^2+λ_2^2\le \bigg( 1-\frac{1}{ω(G)}\bigg)2m\] 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.