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We show that $${\\rm reg}(I(G)^s)\\leq 2s+\\min-match_{\\{K_2, C_5\\}}(G)-1,$$for all $s\\geq 1$, which implies that$${\\rm reg}(I(G)^s)\\leq 2s+\\min-match(G)-1.$$Moreover, we show that$${\\rm reg}(I(G)^s)\\geq 2s+\\ind-match_{\\{K_2, C_5\\}}(G)-2,$$and if $\\ind-match_{\\{K_2, C_5\\}}(G)$ is an odd integer, then$${\\rm reg}(I(G)^s)\\geq 2s+\\ind-match_{\\{K_2, C_5\\}}(G)-1.$$Furthermore, it is shown that$${\\rm reg}(I(G)^s)\\leq 2s+\\ord-match(G)-1,$$where $\\ord-match("},"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":"1805.12508","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-05-30T12:37:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"94d973eb350ec7081d0814e0872becc7d95c0d8d7df407e5243e257728b518e7","abstract_canon_sha256":"943e0b371b137f7badefe1926e7281a9eaab350d08df3aa9fdcd62992d4d3a24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:30.394762Z","signature_b64":"S+OFYBomxL1MNUY1tlkyRDto+jUacZl3K3mIDkeqandrMnJ7eCWukoKSyXwjwfoMCQA+40U07Ls6M6rNQo5MCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab1a3be9babb6bdf535b5d17f3c2072ceb338953804650a6c49b0ae33ce34610","last_reissued_at":"2026-05-17T23:46:30.394217Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:30.394217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved bounds for the regularity of powers of edge ideals of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Seyed Amin Seyed Fakhari, Siamak Yassemi","submitted_at":"2018-05-30T12:37:57Z","abstract_excerpt":"Let $G$ be a graph with edge ideal $I(G)$. We recall the notions of $\\min-match_{\\{K_2, C_5\\}}(G)$ and $\\ind-match_{\\{K_2, C_5\\}}(G)$ from \\cite{sy}. We show that $${\\rm reg}(I(G)^s)\\leq 2s+\\min-match_{\\{K_2, C_5\\}}(G)-1,$$for all $s\\geq 1$, which implies that$${\\rm reg}(I(G)^s)\\leq 2s+\\min-match(G)-1.$$Moreover, we show that$${\\rm reg}(I(G)^s)\\geq 2s+\\ind-match_{\\{K_2, C_5\\}}(G)-2,$$and if $\\ind-match_{\\{K_2, C_5\\}}(G)$ is an odd integer, then$${\\rm reg}(I(G)^s)\\geq 2s+\\ind-match_{\\{K_2, C_5\\}}(G)-1.$$Furthermore, it is shown that$${\\rm reg}(I(G)^s)\\leq 2s+\\ord-match(G)-1,$$where $\\ord-match("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.12508","kind":"arxiv","version":3},"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":"1805.12508","created_at":"2026-05-17T23:46:30.394306+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.12508v3","created_at":"2026-05-17T23:46:30.394306+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.12508","created_at":"2026-05-17T23:46:30.394306+00:00"},{"alias_kind":"pith_short_12","alias_value":"VMNDX2N2XNV5","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_16","alias_value":"VMNDX2N2XNV56U23","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_8","alias_value":"VMNDX2N2","created_at":"2026-05-18T12:32:59.047623+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/VMNDX2N2XNV56U23LUL7HQQHFT","json":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT.json","graph_json":"https://pith.science/api/pith-number/VMNDX2N2XNV56U23LUL7HQQHFT/graph.json","events_json":"https://pith.science/api/pith-number/VMNDX2N2XNV56U23LUL7HQQHFT/events.json","paper":"https://pith.science/paper/VMNDX2N2"},"agent_actions":{"view_html":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT","download_json":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT.json","view_paper":"https://pith.science/paper/VMNDX2N2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.12508&json=true","fetch_graph":"https://pith.science/api/pith-number/VMNDX2N2XNV56U23LUL7HQQHFT/graph.json","fetch_events":"https://pith.science/api/pith-number/VMNDX2N2XNV56U23LUL7HQQHFT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT/action/storage_attestation","attest_author":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT/action/author_attestation","sign_citation":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT/action/citation_signature","submit_replication":"https://pith.science/pith/VMNDX2N2XNV56U23LUL7HQQHFT/action/replication_record"}},"created_at":"2026-05-17T23:46:30.394306+00:00","updated_at":"2026-05-17T23:46:30.394306+00:00"}