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The $D_{i,j} ({\\cal T})$ are called $2$-weights of ${\\cal T}$ and, if we put in order the $2$-weights, the vector which has the $D_{i,j} ({\\cal T})$ as components is called \\emph{$2$-dissimilarity vector} of $ {\\cal T}$. Given a family of positive real numbers $\\{D_{i,j}\\}_{i,j \\in \\{1,...,n\\}}$, we say that a positive-weighted tree ${\\cal T}=(T,w)$ realizes the family if $\\{1,...,n\\} \\subset V(T)$ a"},"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":"1407.0048","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-30T20:27:42Z","cross_cats_sorted":[],"title_canon_sha256":"e6bed242abca741abdd9b8e3f57d6febfdf4864a237d64986cecfbe9a1d613de","abstract_canon_sha256":"1870d147a88d2713d5f8e357ef5516bdb6d152c5c52df47fcf4caa68468cf5de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:35.371301Z","signature_b64":"/Cs7Wt9uodKWeHIiGj/6fQAJQVDfD7pJdeC8OEAkIIeNfMbotVeeTr9A6YP+BP8BglTQdJ+4IW+U4ONXdCFgAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bceabe3f072de1fb0ab628f14ff2570db9851304e8850dde35f322ce07a5ace8","last_reissued_at":"2026-05-18T02:48:35.370725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:35.370725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Buneman's theorem for trees with exatcly n vertices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Agnese Baldisserri","submitted_at":"2014-06-30T20:27:42Z","abstract_excerpt":"Let ${\\cal T}=(T,w)$ be a positive-weighted tree with at least $n$ vertices. For any $i,j \\in \\{1,...,n\\}$, let $D_{i,j} ({\\cal T})$ be the weight of the unique path in $T$ connecting $i$ and $j$. The $D_{i,j} ({\\cal T})$ are called $2$-weights of ${\\cal T}$ and, if we put in order the $2$-weights, the vector which has the $D_{i,j} ({\\cal T})$ as components is called \\emph{$2$-dissimilarity vector} of $ {\\cal T}$. Given a family of positive real numbers $\\{D_{i,j}\\}_{i,j \\in \\{1,...,n\\}}$, we say that a positive-weighted tree ${\\cal T}=(T,w)$ realizes the family if $\\{1,...,n\\} \\subset V(T)$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0048","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":"1407.0048","created_at":"2026-05-18T02:48:35.370811+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.0048v1","created_at":"2026-05-18T02:48:35.370811+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.0048","created_at":"2026-05-18T02:48:35.370811+00:00"},{"alias_kind":"pith_short_12","alias_value":"XTVL4PYHFXQ7","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_16","alias_value":"XTVL4PYHFXQ7WCVW","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_8","alias_value":"XTVL4PYH","created_at":"2026-05-18T12:28:57.508820+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/XTVL4PYHFXQ7WCVWFDYU74SXBW","json":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW.json","graph_json":"https://pith.science/api/pith-number/XTVL4PYHFXQ7WCVWFDYU74SXBW/graph.json","events_json":"https://pith.science/api/pith-number/XTVL4PYHFXQ7WCVWFDYU74SXBW/events.json","paper":"https://pith.science/paper/XTVL4PYH"},"agent_actions":{"view_html":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW","download_json":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW.json","view_paper":"https://pith.science/paper/XTVL4PYH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.0048&json=true","fetch_graph":"https://pith.science/api/pith-number/XTVL4PYHFXQ7WCVWFDYU74SXBW/graph.json","fetch_events":"https://pith.science/api/pith-number/XTVL4PYHFXQ7WCVWFDYU74SXBW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW/action/storage_attestation","attest_author":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW/action/author_attestation","sign_citation":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW/action/citation_signature","submit_replication":"https://pith.science/pith/XTVL4PYHFXQ7WCVWFDYU74SXBW/action/replication_record"}},"created_at":"2026-05-18T02:48:35.370811+00:00","updated_at":"2026-05-18T02:48:35.370811+00:00"}