  Bollob\'as and Nikiforov conjectured that for any graph $G \neq K_n$ with $m$
edges \[ \lambda_1^2+\lambda_2^2\le \bigg( 1-\frac{1}{\omega(G)}\bigg)2m\]
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
$O(m^{1.5-\varepsilon})$ triangles for some $\varepsilon > 0$. In particular,
we show that the conjecture is true for planar graphs, book-free graphs and
cycle-free graphs.
