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For $n\\geq 2k+1$, let $n\\equiv 0,k+1,k+2,\\dots, n-1(\\mod(2k+1))$. Then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a cycle on $n$ vertices and tight bounds otherwise, in terms of $n$ and $k$. We also compute lower bounds for the Stanley depth of the edge id"},"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":"1710.05996","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-10-16T20:45:06Z","cross_cats_sorted":[],"title_canon_sha256":"1acfaeee43485cb082a0a931b116a4602b9b7ea0ee5ff8e5744d19a2f11f373d","abstract_canon_sha256":"95a12d3c49d1a34d0bdc48d80eeb50de4e9ea7c0fcf41170c0165ce2c6b035fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:38.074710Z","signature_b64":"BCoi68gNY6DpbVxRKd62v350l0PTswECFSBS+XS0sS2xJis/S2Bv5trcCKGlt0/GDDw1wv9o+ibjEAbMk6wPCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89d3ab04920301b3928411fcbec7d1e0048f7ff6904af6ef6d59ea4f04f626dc","last_reissued_at":"2026-05-18T00:32:38.074043Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:38.074043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Depth and Stanley depth of the edge ideals of the powers of paths and cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Muhammad Ishaq, Zahid Iqbal","submitted_at":"2017-10-16T20:45:06Z","abstract_excerpt":"Let $k$ be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a path on $n$ vertices. We show that both depth and Stanley depth have the same values and can be given in terms of $k$ and $n$. For $n\\geq 2k+1$, let $n\\equiv 0,k+1,k+2,\\dots, n-1(\\mod(2k+1))$. Then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a cycle on $n$ vertices and tight bounds otherwise, in terms of $n$ and $k$. 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