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 (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 \(\mu _m = d_m-m+2\). In particular we give a full classification of graphs with \(\mu _m = d_m-m+2 \leqslant 1\).
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References
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G.R.W.G. was at Tohoku University while this work took place and was supported by JSPS KAKENHI; Grant Number: 26.03903.
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Greaves, G.R.W., Munemasa, A. & Peng, A. On a Lower Bound for the Laplacian Eigenvalues of a Graph. Graphs and Combinatorics 33, 1509–1519 (2017). https://doi.org/10.1007/s00373-017-1835-y
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DOI: https://doi.org/10.1007/s00373-017-1835-y