  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 $\mu_m =
d_m-m+2 \leqslant 1$.
