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

Abstract

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 < alpha <= 1 or 2 <= alpha < 3, the inequality q(1)(alpha) + ... + q(n)(alpha) >= 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.