A Sharp Lower Bound on the Least Signless Laplacian Eigenvalue of a Graph

Let G be a simple graph with n vertices and m edges, and let $$q_{n}(G)$$ be the least signless Laplacian eigenvalue of G. Recently, Guo, Chen and Yu proved that if $$n\ge 6$$ , then $$\begin{aligned} q_{n}(G)\ge \frac{2m}{n-2}-n+1, \end{aligned}$$ which confirms a conjecture proposed by de Lima, Oliveira, de Abreu and Nikiforov. In this note, we present a new proof for the above inequality, where we also completely characterize the case when the equality holds.