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Moreover, we prove that $ \\operatorname{reg}(I(G)^q)=2q+\\operatorname{reg}(I(G))-2$, for all $ q\\geq 1 $, when $ G $ is a dumbbell graph with a connecting path having no more than two vertices."},"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":"1802.07202","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-20T17:04:32Z","cross_cats_sorted":[],"title_canon_sha256":"3a7c0659dcebc2e10b2b32e6c54285a5cf8f47e52a2cbecf1053fd2ea0070ebc","abstract_canon_sha256":"503ac18b083ffb0d940ada0339a87195dca9a885b786b8bcc311bc126a1e4504"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:00.085215Z","signature_b64":"cKKna0KJJnDs94IHevIc0fctUDkO/3mcHbXpYrRZkNeS+6i9zDQKGda97VFogS68dzbyLoUJOGbL5upFBWHFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"510e4dfa1ab2dda228060fae3619f0952c2130cc12814361865e3b7b393d3ae5","last_reissued_at":"2026-05-18T00:03:00.084749Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:00.084749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of bicyclic Graphs and their powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Beatrice Picone, Navid Nemati, Sepehr Jafari, Yairon Cid-Ruiz","submitted_at":"2018-02-20T17:04:32Z","abstract_excerpt":"Let $I(G)$ be the edge ideal of a bicyclic graph. 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