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

Let G be a simple connected graph on n vertices and m edges. Lima et al. (2011) in [2] posed the following conjecture on the least eigenvalue qn(G)qn(G) of the signless Laplacian of G  : qn(G)⩾2m/(n−1)−n+2qn(G)⩾2m/(n−1)−n+2. In this paper we prove a stronger result: For any graph with n vertices and m   edges, we have qn(G)⩾2m/(n−2)−n+1(n⩾6)qn(G)⩾2m/(n−2)−n+1(n⩾6).