Cvetković–Rowlinson–Simić conjecture on the signless Laplacian spread

Let $s_Q(G)$ denote the difference between the largest and least eigenvalues of the signless Laplacian of a graph $G$. Let $K_{n-1}^+$ be the graph obtained by adding a pendant edge to the complete graph $K_{n-1}$; let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices, respectively.

Cvetković–Rowlinson–Simić conjecture. Over all connected graphs on $n\geq6$ vertices, $s_Q(G)$ is maximized by $K_{n-1}^+$ and minimized by $P_n$ and, when $n$ is odd, by $C_n$.

The minimizing assertion was proved independently by Das and by Fan et al. The maximizing assertion was confirmed in this paper for sufficiently large $n$.