A note on two conjectures relating the independence number and spectral radius of the signless Laplacian matrix of a graph

Abstract

Let G be a simple graph. In this paper, we disprove two conjectures proposed by P. Hansen and C. Lucas in the paper Bounds and conjectures for the signless Laplacian index of graphs. We find an infinite class of graphs as a counterexample for two conjectures relating the spectral radius of the signless Laplacian and the independence number of G.