On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs

Let G be a graph of order n with signless Laplacian eigenvalues q(1),...,q(n) and Laplacian eigenvalues mu(1),...,mu(n). It is proved that for any real number alpha with 0 = mu(alpha)(1) + ... + mu(alpha)(n) holds, and for any real number beta with 1 < beta < 2, the inequality q(1)(beta) + ... + q(n)(beta) <= mu(beta)(1) + ... + mu(beta)(n) holds. In both inequalities, the equality is attained (for alpha is not an element of {1,2}) if and only if G is bipartite.X118sciescopu

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

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 = mu(alpha)(1) + ... + mu(alpha)(n).X11910sciescopu