Sharp Bounds for the Signless Laplacian Spectral Radius in Terms of Clique Number

摘要

In this paper, we present a sharp upper and lower bounds for the signlessLaplacian spectral radius of graphs in terms of clique number. Moreover, theextremal graphs which attain the upper and lower bounds are characterized. Inaddition, these results disprove the two conjectures on the signless Laplacianspectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for thesignless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].