On a lower bound for the Laplacian eigenvalues of a graph

Abstract

If μm and dm denote, respectively, the m-th largest Laplacian eigenvalue and the m-th largest vertex degree of a graph, then 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 μm=dm−m+2. In particular we give a full classification of graphs with μm=dm−m+2⩽1.