Bounds for the largest two eigenvalues of the signless Laplacian

In the paper, a new upper bound for the largest eigenvalue $q_1$ of the signless Laplacian $Q_G=D_G+A_G$ of a graph $G$, generalizing and improving the known bound $q_1\le\Delta_1+\Delta_2$, where $\Delta_1\ge\cdots\ge\Delta_n$ are the ordered vertex degrees, and new lower bounds for the second largest eigenvalue $q_2$ of $Q_G$ are proved. As implications, an upper bound for the difference $q_1-\mu_1$ of the largest eigenvalues of the signless Laplacian $Q_G$ and of the Laplacian $L_G=D_G-A_G$, an upper bound for the largest eigenvalue of the adjacency matrix $A_G$, and an upper bound for the difference $q_1-q_2$ are obtained. All the bounds suggested are expressed in terms of the vertex degrees.