Signless Laplacian energy and spectral radius of a graph

For a graph $G$, the signless Laplacian energy is defined as $E^Q\left(G\right)={\sum }_{i=1}^n\left(q_i-\frac{2m}{n}\right)$, where $n$ and $m$ are the order and the size, and $q_i$ is the $i$th largest signless Laplacian eigenvalue of $G$, respectively. In this paper, we establish a sharp lower bound of the signless Laplacian spectral radius of a graph $G$ in terms of the degree and the average 2-degree, generalizing a well-known lower bound that $q_1\ge \Delta +1$. As applications, we give sharp lower and upper bounds of its signless Laplacian energy for a connected graph and a bipartite graph. All extremal graphs involved are characterized.