Lima et al.'s lower-bound conjecture for the least signless Laplacian eigenvalue

Let $G$ be a connected graph on $n$ vertices and $m$ edges. Let $Q(G)=D(G)+A(G)$ be its signless Laplacian matrix, and write its eigenvalues in nonincreasing order as $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)\geq 0$.

Lima et al.'s conjecture.

$$q_n(G)\geq \frac{2m}{n-1}-n+2.$$

This conjecture concerns a lower bound for the least signless Laplacian eigenvalue of a connected graph. The paper states a stronger result, namely $q_n(G)\geq \frac{2m}{n-2}-n+1$ for any graph with $n\geq 6$, so the conjectured bound is solved.