{"paper":{"title":"On a conjecture for the signless Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jieshan Yang, Lihua You","submitted_at":"2013-06-01T10:37:03Z","abstract_excerpt":"Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\\geq q_2(G)\\geq\\cdots\\geq q_n(G)\\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \\ldots, n.$ F. Ashraf et al. conjectured that $S_k^+(G)\\leq e(G)+\\binom{k+1}{2}$ for $k=1, 2, \\ldots, n.$ In this paper, we give various upper bounds for $S_k^+(G),$ and prove that this conjecture is true for the following cases: connected graph with sufficiently large $k,$ unicyclic graphs and bicyclic graphs for all $k,$ and tricyclic graphs when $k\\neq 3.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0093","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"}