Hansen–Lucas conjectures on the signless Laplacian spectral radius

Let $G$ be a connected graph on $n\ge 4$ vertices. Let $q_1$ be the largest eigenvalue of its signless Laplacian matrix, called the signless Laplacian spectral radius, and let $\omega$ be its clique number.

Hansen–Lucas conjectures. The following inequalities and equality characterizations hold:

$$q_1-\omega\le \frac{3}{2}n-4\quad\text{if }n\text{ is even},$$ $$\frac{q_1}{\omega}\le \frac{n}{2}.$$

The first bound is attained if and only if $n\ge 6$ is even and $G$ is the complement of a perfect matching. When $n\ge 9$ is odd, $q_1-\omega$ is maximized if and only if $G$ is the disjoint union of a complement of a perfect matching on $n-3$ vertices and a triangle on the three remaining vertices. The second bound is attained if and only if $G$ is a complete bipartite graph $K_{p,q}$.

These conjectures give sharp upper bounds for the signless Laplacian spectral radius in terms of clique number, including extremal graph characterizations for both inequalities.