Establish whether the complement of the projective rank is an entanglement-assisted spectral point

Determine whether the graph parameter \(G\mapsto \xi_f(\overline G)\), the projective rank of the complement, belongs to the entanglement-assisted asymptotic spectrum of graphs.

Background

The paper proves the upper bound $\Sigma(G)\le \xi_f(\overline G)$ and constructs a 1440-vertex graph for which the inequality is strict. Thus, the projective-rank parameter cannot be identified with the order-two quantity $\Sigma$ on every graph.

Nevertheless, strict separation from $\Sigma$ does not determine whether the projective-rank parameter satisfies the additivity, multiplicativity, normalization, and entanglement-assisted monotonicity required of a spectral point.