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

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 signless Laplacian 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 a connected graph $G$ on $n$ vertices having a connectivity less than or equal to $k$, respectively, and propose some open problems for future research.