A conjecture on square roots of Laplacian and signless Laplacian eigenvalues of graphs

TL;DR

This paper proposes a conjecture relating the sums of square roots of Laplacian and signless Laplacian eigenvalues of graphs, suggesting a fundamental spectral inequality.

Contribution

It introduces a new conjecture connecting Laplacian and signless Laplacian spectra, offering a potential unifying spectral bound in graph theory.

Findings

Conjecture posited on spectral sums of Laplacian and signless Laplacian eigenvalues.

Provides theoretical motivation and evidence for the conjecture.

Lays groundwork for future proofs or counterexamples.

Abstract

We conjecture that the sum of the square roots of the Laplacian eigenvalues of a graph does not exceed the sum of the square roots of its signless Laplacian eigenvalues.