Elphick–Linz–Wocjan square-sum generalization of the Bollobás–Nikiforov conjecture

Let $G$ be a simple graph on $n$ vertices with adjacency matrix $A(G)$, eigenvalues $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n$, clique number $\omega(G)$, and $n^+$ positive eigenvalues. Define

$$s_k(G)=\sum_{i=1}^k\lambda_i^2,\qquad \Lambda_k(G)=\frac{s_k(G)}{m},$$

where $m$ is the number of edges. Set

$$\ell=\min\{n^+,\omega(G)\}.$$

Elphick–Linz–Wocjan conjecture. For every graph $G$,

$$\Lambda_\ell(G)\leq 2\left(1-\frac{1}{\omega(G)}\right).$$

This generalizes the Bollobás–Nikiforov conjecture and was proposed after computational investigation. The paper verifies it for graphs with at most $O(m^{1.5-\varepsilon})$ triangles for some $\varepsilon>0$, including planar, book-free, and cycle-free graphs; the general statement remains open.