On two conjectures on sum of the powers of signless Laplacian eigenvalues of a graph

Abstract

Let $G$ be a simple graph and $Q(G)$ be the signless Laplacian matrix of $G$. Let $S_\alpha(G)$ be the sum of the $\alpha$-th powers of the nonzero eigenvalues of $Q(G)$. We disprove two conjectures by You and Yang on the extremal values of $S_\alpha(G)$ among bipartite graphs and among graphs with bounded connectivity.