New upper and lower bound for the signless Laplacian spectral radius

Let D be the degree diagonal matrix of G, A be the adjacency matrix of G, Q=D+A be the signless Laplacian matrix of G. Let ξ(G) be the signless Laplacian spectral radius of G. Here the degree of graph was extended to k-degree, and average degree to k-average degree of a graph. A new upper and a new lower bound for the signless spectral radius of a graph G was obtained. Comparisons were made of the result with several classical results on the ξ(G).