{"paper":{"title":"Nordhaus--Gaddum type inequalities for Laplacian and signless Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"B. Tayfeh-Rezaie, F. Ashraf","submitted_at":"2014-02-12T22:08:12Z","abstract_excerpt":"Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $\\mu_1(G)\\ge\\cdots\\ge\\mu_{n-1}(G)\\ge\\mu_n(G)=0$. It is a conjecture on Laplacian spread of graphs that $\\mu_1(G)-\\mu_{n-1}(G)\\le n-1$ or equivalently $\\mu_1(G)+\\mu_1(\\Gb)\\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $\\mu_1(G)\\mu_1(\\Gb)\\le n(n-1)$. Aouchiche and Hansen [A survey of Nordhaus--Gaddum type relations, Discrete Appl. Math. 161 (2013), 466--546] conjectured that %for any graph $G$ with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.2995","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"}