{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:WDMBDXUTP3HWPQJ5USRAL5NYDA","short_pith_number":"pith:WDMBDXUT","schema_version":"1.0","canonical_sha256":"b0d811de937ecf67c13da4a205f5b8182dc1488864126cd68dcff76c0aa82065","source":{"kind":"arxiv","id":"1210.7004","version":1},"attestation_state":"computed","paper":{"title":"The inverse inertia problem for the complements of partial $k$-trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hein van der Holst","submitted_at":"2012-10-25T21:48:17Z","abstract_excerpt":"Let $\\mathbb{F}$ be an infinite field with characteristic different from two. For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;\\mathbb{F})$ be the set of all symmetric $n\\times n$ matrices $A=[a_{i,j}]$ over $\\mathbb{F}$ with $a_{i,j}\\not=0$, $i\\not=j$ if and only if $ij\\in E$. We show that if $G$ is the complement of a partial $k$-tree and $m\\geq k+2$, then for all nonsingular symmetric $m\\times m$ matrices $K$ over $\\mathbb{F}$, there exists an $m\\times n$ matrix $U$ such that $U^T K U\\in S(G;\\mathbb{F})$. As a corollary we obtain that, if $k+2\\leq m\\leq n$ and $G$ is the complement of a p"},"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":"1210.7004","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-25T21:48:17Z","cross_cats_sorted":[],"title_canon_sha256":"53b80750a7096a755ea0668974a4b535291f522f441a6ac12c6ca1d7b6aa3e21","abstract_canon_sha256":"3db07542b98ae1265024ec204b62054528c98babc2b96841cd5c22762035a901"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:22.069214Z","signature_b64":"WBowqsmJF02baSXES8BWeFefckkzV09quaHndntOe0zlXZnbvbB9sJu9YYmNMDcuo6mAbFtfWk6IWACAZJoPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0d811de937ecf67c13da4a205f5b8182dc1488864126cd68dcff76c0aa82065","last_reissued_at":"2026-05-18T03:42:22.067995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:22.067995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The inverse inertia problem for the complements of partial $k$-trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hein van der Holst","submitted_at":"2012-10-25T21:48:17Z","abstract_excerpt":"Let $\\mathbb{F}$ be an infinite field with characteristic different from two. For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;\\mathbb{F})$ be the set of all symmetric $n\\times n$ matrices $A=[a_{i,j}]$ over $\\mathbb{F}$ with $a_{i,j}\\not=0$, $i\\not=j$ if and only if $ij\\in E$. We show that if $G$ is the complement of a partial $k$-tree and $m\\geq k+2$, then for all nonsingular symmetric $m\\times m$ matrices $K$ over $\\mathbb{F}$, there exists an $m\\times n$ matrix $U$ such that $U^T K U\\in S(G;\\mathbb{F})$. As a corollary we obtain that, if $k+2\\leq m\\leq n$ and $G$ is the complement of a p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.7004","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":"1210.7004","created_at":"2026-05-18T03:42:22.068125+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.7004v1","created_at":"2026-05-18T03:42:22.068125+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.7004","created_at":"2026-05-18T03:42:22.068125+00:00"},{"alias_kind":"pith_short_12","alias_value":"WDMBDXUTP3HW","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"WDMBDXUTP3HWPQJ5","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"WDMBDXUT","created_at":"2026-05-18T12:27:25.539911+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/WDMBDXUTP3HWPQJ5USRAL5NYDA","json":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA.json","graph_json":"https://pith.science/api/pith-number/WDMBDXUTP3HWPQJ5USRAL5NYDA/graph.json","events_json":"https://pith.science/api/pith-number/WDMBDXUTP3HWPQJ5USRAL5NYDA/events.json","paper":"https://pith.science/paper/WDMBDXUT"},"agent_actions":{"view_html":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA","download_json":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA.json","view_paper":"https://pith.science/paper/WDMBDXUT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.7004&json=true","fetch_graph":"https://pith.science/api/pith-number/WDMBDXUTP3HWPQJ5USRAL5NYDA/graph.json","fetch_events":"https://pith.science/api/pith-number/WDMBDXUTP3HWPQJ5USRAL5NYDA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA/action/storage_attestation","attest_author":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA/action/author_attestation","sign_citation":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA/action/citation_signature","submit_replication":"https://pith.science/pith/WDMBDXUTP3HWPQJ5USRAL5NYDA/action/replication_record"}},"created_at":"2026-05-18T03:42:22.068125+00:00","updated_at":"2026-05-18T03:42:22.068125+00:00"}