On a lower bound for the Laplacian eigenvalues of a graph

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 $\mu_m = d_m-m+2$\mathrm{.}Inparticularwegiveafullclassificationofgraphswith$\mu_m = d_m-m+2 \leqslant 1$.