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For a graph $G$ and a non-zero real number $\alpha$, the graph invariant$S_{\alpha}(G)$ is the sum of the $\alpha^{th}$ power of the non-zero signlessLaplacian eigenvalues of $G$. In this paper, we obtain the sharp bounds of$S_{\alpha}(G)$ for a connected bipartite graph $G$ on $n$ vertices and aconnected graph $G$ on $n$ vertices having a connectivity less than or equal to$k$, respectively, and propose some open problems for future research.

Notes on the sum of powers of the signless Laplacian eigenvalues of graphs