On a lower bound for the Laplacian eigenvalues of a graph

TL;DR

This paper investigates the characterization of graphs that meet a specific lower bound for Laplacian eigenvalues, providing a full classification for cases where the bound is less than or equal to one.

Contribution

It offers a complete classification of graphs satisfying the equality _m = d_m - m + 2  for the case when this value is at most one.

Findings

Characterization of graphs with _m = d_m - m + 2  0 

Complete classification for graphs where the bound is 0 1

Extension of the conjecture by Guo and proof by Brouwer and Haemers

Abstract

If $\mu_m$ and $d_m$ denote, respectively, the $m$-th largest Laplacian eigenvalue and the $m$-th largest vertex degree of a graph, then $\mu_m \geqslant d_m-m+2$. This inequality was conjectured by Guo in 2007 and proved by Brouwer and Haemers in 2008. Brouwer and Haemers gave several examples of graphs achieving equality, but a complete characterisation was not given. In this paper we consider the problem of characterising graphs satisfying $\mu_m = d_m-m+2$. In particular we give a full classification of graphs with $\mu_m = d_m-m+2 \leqslant 1$.