On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs (Q1960267)

Let \(G\) be a graph of order \(n\) with signless Laplacian eigenvalues \(q_{1},\dots ,q_{n}\) and Laplacian eigenvalues \(\mu_{1},\dots ,\mu_{n}\). It is proved that for any real number \(\alpha\) with \(0<\alpha \leq 1\) or \(2\leq \alpha <3\), the inequality \(q_{1}^\alpha+\dots + q_{n}^\alpha\geq \mu_{1}^\alpha+\dots + \mu_{n}^\alpha\) holds, and for any real number \(\beta \) with \(1<\beta<2\), the inequality \(q_{1}^\beta +\dots + q_{n}^\beta\geq \mu_{1}^\beta +\dots + \mu_{n}^\beta\) holds. In both inequalities, the equality is attained (for \(\alpha \notin \{1,2\})\) if and only if \(G\) is bipartite.