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

TL;DR

This paper investigates bounds on the sum of powers of signless Laplacian eigenvalues in graphs, providing sharp bounds for specific graph classes and suggesting directions for future research.

Contribution

It derives sharp bounds for the invariant $S_{\alpha}(G)$ in connected bipartite graphs and graphs with limited connectivity, advancing spectral graph theory understanding.

Findings

Established sharp bounds for $S_{\alpha}(G)$ in bipartite graphs.

Derived bounds for graphs with connectivity up to $k$.

Proposed open problems for further exploration.

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 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.