Generalising a conjecture due to Bollobas and Nikiforov

Let $\mu_1 \ge \ldots \ge \mu_n$ denote the eigenvalues of a graph $G$ with $m$ edges and clique number $\omega(G)$. Nikiforov proved a spectral version of Tur\'an's theorem that \[ \mu_1^2 \le \frac{2m(\omega - 1)}{\omega}, \] and Bollob\'as and Nikiforov conjectured that for $G \not = K_n$ \[ \mu_1^2 + \mu_2^2 \le \frac{2m(\omega - 1)}{\omega}. \] This paper proposes the conjecture that for all graphs $(\mu_1^2 + \mu_2^2)$ in this inequality can be replaced by the sum of the squares of the $\omega$ largest eigenvalues, provided they are positive. We provide experimental and theoretical evidence for this conjecture, and describe how the bound can be applied.