{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:6SCZTPOYGNZTCWGXO24BONT4BF","short_pith_number":"pith:6SCZTPOY","schema_version":"1.0","canonical_sha256":"f48599bdd833733158d776b817367c097786fd663c9b53e594960a5e32de168c","source":{"kind":"arxiv","id":"1908.09202","version":1},"attestation_state":"computed","paper":{"title":"Wiener indices of maximal $k$-degenerate graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allan Bickle, Zhongyuan Che","submitted_at":"2019-08-24T20:41:03Z","abstract_excerpt":"A graph is maximal $k$-degenerate if each induced subgraph has a vertex of degree at most $k$ and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal $k$-degenerate graphs of order $n \\ge k \\ge 1$. A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal $k$-degenerate graphs of order $n \\ge k$ are $k$-trees. For $k$-trees of order $n \\ge 2k+2$, we characterize all extremal graphs for the upper bound."},"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":"1908.09202","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-24T20:41:03Z","cross_cats_sorted":[],"title_canon_sha256":"8d558fe90c0b739df776bea608be88227d6b664e186a5e2cbc463a4adb45e918","abstract_canon_sha256":"7ba888d4c7104aeae128c8e2ba1a7068aee1591c51591c8c50efbe526cab3812"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:59:37.604121Z","signature_b64":"qxrFJOG6PZ1Fn9lfpx9waSMQwATqDVMkkUa8xxn1L8IEE6NjQwxsW3MBzrtcayqiz1lbyRsh5uRpgBkQB+dhBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f48599bdd833733158d776b817367c097786fd663c9b53e594960a5e32de168c","last_reissued_at":"2026-07-04T23:59:37.603708Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:59:37.603708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Wiener indices of maximal $k$-degenerate graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allan Bickle, Zhongyuan Che","submitted_at":"2019-08-24T20:41:03Z","abstract_excerpt":"A graph is maximal $k$-degenerate if each induced subgraph has a vertex of degree at most $k$ and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal $k$-degenerate graphs of order $n \\ge k \\ge 1$. A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal $k$-degenerate graphs of order $n \\ge k$ are $k$-trees. For $k$-trees of order $n \\ge 2k+2$, we characterize all extremal graphs for the upper bound."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.09202","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.09202/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.09202","created_at":"2026-07-04T23:59:37.603767+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.09202v1","created_at":"2026-07-04T23:59:37.603767+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.09202","created_at":"2026-07-04T23:59:37.603767+00:00"},{"alias_kind":"pith_short_12","alias_value":"6SCZTPOYGNZT","created_at":"2026-07-04T23:59:37.603767+00:00"},{"alias_kind":"pith_short_16","alias_value":"6SCZTPOYGNZTCWGX","created_at":"2026-07-04T23:59:37.603767+00:00"},{"alias_kind":"pith_short_8","alias_value":"6SCZTPOY","created_at":"2026-07-04T23:59:37.603767+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/6SCZTPOYGNZTCWGXO24BONT4BF","json":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF.json","graph_json":"https://pith.science/api/pith-number/6SCZTPOYGNZTCWGXO24BONT4BF/graph.json","events_json":"https://pith.science/api/pith-number/6SCZTPOYGNZTCWGXO24BONT4BF/events.json","paper":"https://pith.science/paper/6SCZTPOY"},"agent_actions":{"view_html":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF","download_json":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF.json","view_paper":"https://pith.science/paper/6SCZTPOY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.09202&json=true","fetch_graph":"https://pith.science/api/pith-number/6SCZTPOYGNZTCWGXO24BONT4BF/graph.json","fetch_events":"https://pith.science/api/pith-number/6SCZTPOYGNZTCWGXO24BONT4BF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF/action/storage_attestation","attest_author":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF/action/author_attestation","sign_citation":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF/action/citation_signature","submit_replication":"https://pith.science/pith/6SCZTPOYGNZTCWGXO24BONT4BF/action/replication_record"}},"created_at":"2026-07-04T23:59:37.603767+00:00","updated_at":"2026-07-04T23:59:37.603767+00:00"}