Notes on the Sum of Powers of the Signless Laplacian Eigenvalues of Graphs

Abstract

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 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.