{"paper":{"title":"Signless Laplacian eigenvalue problems of Nordhaus-Gaddum type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huiqiu Lin, Xueyi Huang","submitted_at":"2019-04-30T13:35:53Z","abstract_excerpt":"Let $G$ be a graph of order $n$, and let $q_1(G)\\geq q_2(G)\\geq\\cdots\\geq q_n(G)$ denote the signless Laplacian eigenvalues of $G$. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) \\#P3.6] showed that $q_1(G)+q_1(\\overline{G})\\leq 3n-4$, with equality holding if and only if $G$ or $\\overline{G}$ is the star $K_{1,n-1}$. In this paper, we discuss the following problem: for $n\\geq6$, does $q_2(G)+q_2(\\overline{G})\\leq 2n-5$ always hold? We provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs, and determine the graphs attaining"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.13225","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"}