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 $q_1, \ldots,q_n$ and Laplacian eigenvalues $\mu_1,\ldots,\mu_n$. It is proved that for any real number $\alpha$ with $0 < \alpha\leq1$ or $2\leq\alpha < 3$, the inequality $q_1^\alpha+\cdots+ q_n^\alpha\geq \mu_1^\alpha+\cdots+\mu_n^\alpha$ holds, and for any real number $\beta$ with $1 < \beta < 2$, the inequality $q_1^\beta+\cdots+ q_n^\beta\le \mu_1^\beta+\cdots+\mu_n^\beta$ holds. In both inequalities, the equality is attained (for $\alpha \notin \{1,2\}$) if and only if $G$ is bipartite.