{"paper":{"title":"Remoteness and distance eigenvalues of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Baoyindureng Wu, Huiqiu Lin, Kinkar Ch. Das","submitted_at":"2015-07-25T09:17:29Z","abstract_excerpt":"Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $\\rho$ of $G$ is the maximum average distance from a vertex to all others and $\\partial_1\\geq\\cdots\\geq \\partial_n$ are the distance eigenvalues of $G$. In \\cite{AH}, Aouchiche and Hansen conjectured that $\\rho+\\partial_3>0$ when $d\\geq 3$ and $\\rho+\\partial_{\\lfloor\\frac{7d}{8}\\rfloor}>0.$ In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on $\\partial_n+\\rho$ and $\\partial_1-\\rho$ when $G\\ncong K_n$ and the extremal graphs are characterized."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07083","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"}