{"paper":{"title":"Graph functions maximized on a path","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Celso Marques da Silva Jr, Vladimir Nikiforov","submitted_at":"2014-12-28T21:30:27Z","abstract_excerpt":"Given a connected graph $G\\ $of order $n$ and a nonnegative symmetric matrix $A=\\left[ a_{i,j}\\right] $ of order $n,$ define the function $F_{A}\\left( G\\right) $ as% \\[ F_{A}\\left( G\\right) =\\sum_{1\\leq i<j\\leq n}d_{G}\\left( i,j\\right) a_{i,j}, \\] where $d_{G}\\left( i,j\\right) $ denotes the distance between the vertices $i$ and $j$ in $G.$\n  In this note it is shown that $F_{A}\\left( G\\right) \\leq F_{A}\\left( P\\right) \\,$for some path of order $n.$ Moreover, if each row of $A$ has at most one zero off-diagonal entry, then $F_{A}\\left( G\\right) <F_{A}\\left( P\\right) \\,$for some path of order $n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8215","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"}