{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:BG5W7UQPUSRSMEP2WJRTWI2FLJ","short_pith_number":"pith:BG5W7UQP","schema_version":"1.0","canonical_sha256":"09bb6fd20fa4a32611fab2633b23455a4d6b73a65498b2067e9a5d87ed29f8fc","source":{"kind":"arxiv","id":"1412.8215","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1412.8215","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-28T21:30:27Z","cross_cats_sorted":[],"title_canon_sha256":"43b9d7fcfcb0e2a933f69493a0a9ded6d514ac5d351f358fe092d085d716ac75","abstract_canon_sha256":"14c558de8424db35799caf87640340afce09c3042ee30fd48e96898e67bf7c87"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:23.768729Z","signature_b64":"i5vrY7Pc8fTGrnxc7rvkR3p/FbSu7sWBOePEf0JIweKxchIUDslJUOMDVBqMoyGZrSmURolPigrhNn4F4AYzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09bb6fd20fa4a32611fab2633b23455a4d6b73a65498b2067e9a5d87ed29f8fc","last_reissued_at":"2026-05-18T02:30:23.768259Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:23.768259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1412.8215","created_at":"2026-05-18T02:30:23.768324+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.8215v1","created_at":"2026-05-18T02:30:23.768324+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8215","created_at":"2026-05-18T02:30:23.768324+00:00"},{"alias_kind":"pith_short_12","alias_value":"BG5W7UQPUSRS","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"BG5W7UQPUSRSMEP2","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"BG5W7UQP","created_at":"2026-05-18T12:28:22.404517+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ","json":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ.json","graph_json":"https://pith.science/api/pith-number/BG5W7UQPUSRSMEP2WJRTWI2FLJ/graph.json","events_json":"https://pith.science/api/pith-number/BG5W7UQPUSRSMEP2WJRTWI2FLJ/events.json","paper":"https://pith.science/paper/BG5W7UQP"},"agent_actions":{"view_html":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ","download_json":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ.json","view_paper":"https://pith.science/paper/BG5W7UQP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.8215&json=true","fetch_graph":"https://pith.science/api/pith-number/BG5W7UQPUSRSMEP2WJRTWI2FLJ/graph.json","fetch_events":"https://pith.science/api/pith-number/BG5W7UQPUSRSMEP2WJRTWI2FLJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ/action/storage_attestation","attest_author":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ/action/author_attestation","sign_citation":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ/action/citation_signature","submit_replication":"https://pith.science/pith/BG5W7UQPUSRSMEP2WJRTWI2FLJ/action/replication_record"}},"created_at":"2026-05-18T02:30:23.768324+00:00","updated_at":"2026-05-18T02:30:23.768324+00:00"}