{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:TDP2SKXXLNHSJM3MCLWXQN3XGE","short_pith_number":"pith:TDP2SKXX","schema_version":"1.0","canonical_sha256":"98dfa92af75b4f24b36c12ed7837773107d554fda4ab12ccaafe6ff5bbe0f755","source":{"kind":"arxiv","id":"2009.10294","version":1},"attestation_state":"computed","paper":{"title":"On the least size of a graph with a given degree set -- II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aditya Sahdev, Amitabha Tripathi, Jai Moondra","submitted_at":"2020-09-22T03:00:02Z","abstract_excerpt":"The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\\mathscr D$ is $1+\\max \\mathscr D$. Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set $\\mathscr D$. We expand on their results, and determine the least size of graphs with degree set $\\mathscr D$ when (i) $\\min \\mathscr D \\mid d$ for each $d \\in \\mathscr D$; (ii) $\\min \\mathscr D=2$; (iii) $\\mathscr D=\\{m,m+1,\\ldots,n\\}$. In addition, given any $\\ma"},"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":"2009.10294","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-22T03:00:02Z","cross_cats_sorted":[],"title_canon_sha256":"de5b09cf706a62869f2e6163456fe3ca1f49a82db0e7454353bef5ca5580e444","abstract_canon_sha256":"eecbeae86f9dbfc5edb131634e0bf14f92962cfcc8c38317b23e7747bbc3047d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:32:37.889712Z","signature_b64":"X5bdT4eReIPei7UGa/jPG3Sn5b0YXRhftBIMPdy/KKBrJKKMCNDM2EAiHTw0niMN2ihK3Mykum+DkhN1Ang1Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98dfa92af75b4f24b36c12ed7837773107d554fda4ab12ccaafe6ff5bbe0f755","last_reissued_at":"2026-07-05T09:32:37.889158Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:32:37.889158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the least size of a graph with a given degree set -- II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aditya Sahdev, Amitabha Tripathi, Jai Moondra","submitted_at":"2020-09-22T03:00:02Z","abstract_excerpt":"The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\\mathscr D$ is $1+\\max \\mathscr D$. Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set $\\mathscr D$. We expand on their results, and determine the least size of graphs with degree set $\\mathscr D$ when (i) $\\min \\mathscr D \\mid d$ for each $d \\in \\mathscr D$; (ii) $\\min \\mathscr D=2$; (iii) $\\mathscr D=\\{m,m+1,\\ldots,n\\}$. In addition, given any $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.10294","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/2009.10294/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":"2009.10294","created_at":"2026-07-05T09:32:37.889222+00:00"},{"alias_kind":"arxiv_version","alias_value":"2009.10294v1","created_at":"2026-07-05T09:32:37.889222+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.10294","created_at":"2026-07-05T09:32:37.889222+00:00"},{"alias_kind":"pith_short_12","alias_value":"TDP2SKXXLNHS","created_at":"2026-07-05T09:32:37.889222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TDP2SKXXLNHSJM3M","created_at":"2026-07-05T09:32:37.889222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TDP2SKXX","created_at":"2026-07-05T09:32:37.889222+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/TDP2SKXXLNHSJM3MCLWXQN3XGE","json":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE.json","graph_json":"https://pith.science/api/pith-number/TDP2SKXXLNHSJM3MCLWXQN3XGE/graph.json","events_json":"https://pith.science/api/pith-number/TDP2SKXXLNHSJM3MCLWXQN3XGE/events.json","paper":"https://pith.science/paper/TDP2SKXX"},"agent_actions":{"view_html":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE","download_json":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE.json","view_paper":"https://pith.science/paper/TDP2SKXX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2009.10294&json=true","fetch_graph":"https://pith.science/api/pith-number/TDP2SKXXLNHSJM3MCLWXQN3XGE/graph.json","fetch_events":"https://pith.science/api/pith-number/TDP2SKXXLNHSJM3MCLWXQN3XGE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE/action/storage_attestation","attest_author":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE/action/author_attestation","sign_citation":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE/action/citation_signature","submit_replication":"https://pith.science/pith/TDP2SKXXLNHSJM3MCLWXQN3XGE/action/replication_record"}},"created_at":"2026-07-05T09:32:37.889222+00:00","updated_at":"2026-07-05T09:32:37.889222+00:00"}