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

Abstract

Let G be a graph of order n such that Sigma(n)(i)=0(-1) (i)a(i)gimel(n-i) and Sigma(n)(i=0)(-1)(i)b(i)gimel(n-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 a, 0 < alpha <= 1, if q(1), ... , q(n) and mu(1), ... ,mu(n) are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then q(1)(alpha) + ... + q(n)(alpha) >= mu(alpha)(1) + ... + mu(alpha)(n).