{"paper":{"title":"Median eigenvalues of bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Behruz Tayfeh-Rezaie, Bojan Mohar","submitted_at":"2013-12-09T21:59:43Z","abstract_excerpt":"For a graph $G$ of order $n$ and with eigenvalues $\\lambda_1\\geqslant\\cdots\\geqslant\\lambda_n$, the HL-index $R(G)$ is defined as $R(G) ={\\max}\\left\\{|\\lambda_{\\lfloor(n+1)/2\\rfloor}|, |\\lambda_{\\lceil(n+1)/2\\rceil}|\\right\\}.$ We show that for every connected bipartite graph $G$ with maximum degree $\\Delta\\geqslant3$, $R(G)\\leqslant\\sqrt{\\Delta-2}$ unless $G$ is the the incidence graph of a projective plane of order $\\Delta-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2613","kind":"arxiv","version":1},"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"}