A Note on the Signless Laplacian and Distance Signless Laplacian Eigenvalues of Graphs

Let G be a simple graph. We first show that δ≥di-√[i/2][i/2], where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ（G）, we determine the extremal graphs with the minimum ρQ（G） among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.