On two conjectures on sum of the powers of signless Laplacian eigenvalues of a graph

TL;DR

This paper disprove two conjectures regarding the extremal values of the sum of powers of signless Laplacian eigenvalues in bipartite graphs and graphs with bounded connectivity.

Contribution

It provides counterexamples to two conjectures about the extremal properties of signless Laplacian eigenvalues in specific graph classes.

Findings

Disproved conjectures on extremal eigenvalue sums.

Identified specific graph classes where conjectures do not hold.

Enhanced understanding of spectral properties of signless Laplacian matrices.

Abstract

Let $G$ be a simple graph and $Q(G)$ be the signless Laplacian matrix of $G$. Let $S_\alpha(G)$ be the sum of the $\alpha$-th powers of the nonzero eigenvalues of $Q(G)$. We disprove two conjectures by You and Yang on the extremal values of $S_\alpha(G)$ among bipartite graphs and among graphs with bounded connectivity.