A relation between the Laplacian and signless Laplacian eigenvalues of a graph

Abstract

Let G be a graph of order n such that ∑ni=0(-1) iailambdan-i and ∑ni=0(-1) iailambdan-i are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i ≥b i for i=0,1,⋯,n. As a consequence, we prove that for any α, 0