A lower bound of the least signless Laplacian eigenvalue of a graph

Let $G$ be a simple connected graph on $n$ vertices and $m$ edges. In [Linear Algebra Appl. 435 (2011) 2570-2584], Lima et al. posed the following conjecture on the least eigenvalue $q_n(G)$ of the signless Laplacian of $G$: $\displaystyle q_n(G)\ge {2m}/{(n-1)}-n+2$. In this paper we prove a stronger result: For any graph with $n$ vertices and $m$ edges, we have $\displaystyle q_n(G)\ge {2m}/{(n-2)}-n+1 (n\ge 6)$.