{"paper":{"title":"On a lower bound for the Laplacian eigenvalues of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akihiro Munemasa, Anni Peng, Gary R. W. Greaves","submitted_at":"2017-07-11T11:20:11Z","abstract_excerpt":"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 $\\mu_m = d_m-m+2 \\leqslant 1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03221","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}