On a Lower Bound for the Laplacian Eigenvalues of a Graph

If $$\mu _m$$μm and $$d_m$$dm denote, respectively, the m-th largest Laplacian eigenvalue and the m-th largest vertex degree of a graph, then $$\mu _m \geqslant d_m-m+2$$μm⩾dm-m+2. This inequality was conjectured by Guo (Linear Multilinear Algebra 55:93–102, 2007) and proved by Brouwer and Haemers (Linear Algebra Appl 429:2131–2135, 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 $$\mu _m = d_m-m+2$$μm=dm-m+2. In particular we give a full classification of graphs with $$\mu _m = d_m-m+2 \leqslant 1$$μm=dm-m+2⩽1.