Sharp bounds on the signless laplacian spectral radii of graphs

Let G be a simple graph with vertices v1,v2,⋯,vn, of degrees Δ=d1⩾d2⩾⋯⩾dn=δ, respectively. Let A be the (0,1)-adjacency matrix of G and D be the diagonal matrix diag(d1,d2,⋯,dn). Q(G)=D+A is called the signless Laplacian of G. In this paper, we give some sharp bounds on signless Laplacian spectral radius ρ(Q(G)) in terms of di.