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We prove that$$k{\\rm deg}(J(G))\\leq {\\rm reg}(J(G)^{(k)})\\leq (k-1){\\rm deg}(J(G))+|V(G)|-1.$$We also determine families of graphs for which the above inequalities are equality."},"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":"1709.03077","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-09-10T09:54:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"43b21fb9ab0ae60b647c46fc179132275248db0666d1c5909c92a9ad1a28ccc3","abstract_canon_sha256":"77054667d205443fee5c5dd2651d8160dab50b06370581180e2fc6ad0cb7e91a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:39.462175Z","signature_b64":"4UyfJpnAfWZhBmX9Bdwb+vUyy77AZscqkR8EDFg7xARAsbz/DuMTB0hflnuuF519O2Q2tKAD8OpMTgJQKEUoCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d33a2381b9da818f60ca5a87e397caa408e0f373ddf947e9977f7453f15bbf2","last_reissued_at":"2026-05-18T00:35:39.461621Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:39.461621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of symbolic powers of cover ideals of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"S. 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We prove that$$k{\\rm deg}(J(G))\\leq {\\rm reg}(J(G)^{(k)})\\leq (k-1){\\rm deg}(J(G))+|V(G)|-1.$$We also determine families of graphs for which the above inequalities are equality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.03077","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":"1709.03077","created_at":"2026-05-18T00:35:39.461723+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.03077v1","created_at":"2026-05-18T00:35:39.461723+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.03077","created_at":"2026-05-18T00:35:39.461723+00:00"},{"alias_kind":"pith_short_12","alias_value":"BUZ2EOA3TWUB","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BUZ2EOA3TWUBR5QM","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BUZ2EOA3","created_at":"2026-05-18T12:31:08.081275+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/BUZ2EOA3TWUBR5QMUWUH4OL4VJ","json":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ.json","graph_json":"https://pith.science/api/pith-number/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/graph.json","events_json":"https://pith.science/api/pith-number/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/events.json","paper":"https://pith.science/paper/BUZ2EOA3"},"agent_actions":{"view_html":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ","download_json":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ.json","view_paper":"https://pith.science/paper/BUZ2EOA3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.03077&json=true","fetch_graph":"https://pith.science/api/pith-number/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/graph.json","fetch_events":"https://pith.science/api/pith-number/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/action/storage_attestation","attest_author":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/action/author_attestation","sign_citation":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/action/citation_signature","submit_replication":"https://pith.science/pith/BUZ2EOA3TWUBR5QMUWUH4OL4VJ/action/replication_record"}},"created_at":"2026-05-18T00:35:39.461723+00:00","updated_at":"2026-05-18T00:35:39.461723+00:00"}