The Grone-Merris Conjecture

TL;DR

This paper provides a complete proof of the Grone-Merris conjecture, which states that the Laplacian spectrum of a finite graph is majorized by its conjugate degree sequence, advancing understanding in spectral graph theory.

Contribution

The paper offers the first complete proof of the longstanding Grone-Merris conjecture in spectral graph theory.

Findings

Confirmed the conjecture for all finite graphs

Established the majorization relation between Laplacian spectrum and conjugate degree sequence

Enhanced theoretical understanding of graph spectra and degree sequences

Abstract

In spectral graph theory, Grone and Merris conjecture that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of this graph. We give a complete proof for this conjecture.