{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:RHJ2WBESAMA3HEUECH6L5R6R4A","short_pith_number":"pith:RHJ2WBES","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"},"canonical_sha256":"89d3ab04920301b3928411fcbec7d1e0048f7ff6904af6ef6d59ea4f04f626dc","source":{"kind":"arxiv","id":"1710.05996","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.05996","created_at":"2026-05-18T00:32:38Z"},{"alias_kind":"arxiv_version","alias_value":"1710.05996v1","created_at":"2026-05-18T00:32:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.05996","created_at":"2026-05-18T00:32:38Z"},{"alias_kind":"pith_short_12","alias_value":"RHJ2WBESAMA3","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RHJ2WBESAMA3HEUE","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RHJ2WBES","created_at":"2026-05-18T12:31:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:RHJ2WBESAMA3HEUECH6L5R6R4A","target":"record","payload":{"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"},"canonical_sha256":"89d3ab04920301b3928411fcbec7d1e0048f7ff6904af6ef6d59ea4f04f626dc","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"},"source_kind":"arxiv","source_id":"1710.05996","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:32:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KSBPMpKefNAR3l/qIbCEA2osGvjY29R1v/sC0k5wKS+4qDNSdZwiloQ/xRXJ0UevBDFaZ6ow1h3zO2VtpA8UDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T14:43:08.857339Z"},"content_sha256":"669a2ac3411f578e9fdf035e5e3cb150b373403e65e06c91b673348b66da5aea","schema_version":"1.0","event_id":"sha256:669a2ac3411f578e9fdf035e5e3cb150b373403e65e06c91b673348b66da5aea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:RHJ2WBESAMA3HEUECH6L5R6R4A","target":"graph","payload":{"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 $k$. We also compute lower bounds for the Stanley depth of the edge id"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05996","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:32:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RfzD2/2rmP6jv+80x7b0KhbeWg81YBis2bDvyeW4YKKjuXQV2lHTXp6bMDAS3Zympe5mYWMWvVv8tyvI+8uiDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T14:43:08.857974Z"},"content_sha256":"9d3d48b2e2c5ba8d4d12f279a2a5ad4e22eb800020941bbae2a3bd711bf99d93","schema_version":"1.0","event_id":"sha256:9d3d48b2e2c5ba8d4d12f279a2a5ad4e22eb800020941bbae2a3bd711bf99d93"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RHJ2WBESAMA3HEUECH6L5R6R4A/bundle.json","state_url":"https://pith.science/pith/RHJ2WBESAMA3HEUECH6L5R6R4A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RHJ2WBESAMA3HEUECH6L5R6R4A/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T14:43:08Z","links":{"resolver":"https://pith.science/pith/RHJ2WBESAMA3HEUECH6L5R6R4A","bundle":"https://pith.science/pith/RHJ2WBESAMA3HEUECH6L5R6R4A/bundle.json","state":"https://pith.science/pith/RHJ2WBESAMA3HEUECH6L5R6R4A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RHJ2WBESAMA3HEUECH6L5R6R4A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:RHJ2WBESAMA3HEUECH6L5R6R4A","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"95a12d3c49d1a34d0bdc48d80eeb50de4e9ea7c0fcf41170c0165ce2c6b035fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-10-16T20:45:06Z","title_canon_sha256":"1acfaeee43485cb082a0a931b116a4602b9b7ea0ee5ff8e5744d19a2f11f373d"},"schema_version":"1.0","source":{"id":"1710.05996","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.05996","created_at":"2026-05-18T00:32:38Z"},{"alias_kind":"arxiv_version","alias_value":"1710.05996v1","created_at":"2026-05-18T00:32:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.05996","created_at":"2026-05-18T00:32:38Z"},{"alias_kind":"pith_short_12","alias_value":"RHJ2WBESAMA3","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"RHJ2WBESAMA3HEUE","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"RHJ2WBES","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:9d3d48b2e2c5ba8d4d12f279a2a5ad4e22eb800020941bbae2a3bd711bf99d93","target":"graph","created_at":"2026-05-18T00:32:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $k$. We also compute lower bounds for the Stanley depth of the edge id","authors_text":"Muhammad Ishaq, Zahid Iqbal","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-10-16T20:45:06Z","title":"Depth and Stanley depth of the edge ideals of the powers of paths and cycles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05996","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:669a2ac3411f578e9fdf035e5e3cb150b373403e65e06c91b673348b66da5aea","target":"record","created_at":"2026-05-18T00:32:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"95a12d3c49d1a34d0bdc48d80eeb50de4e9ea7c0fcf41170c0165ce2c6b035fe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-10-16T20:45:06Z","title_canon_sha256":"1acfaeee43485cb082a0a931b116a4602b9b7ea0ee5ff8e5744d19a2f11f373d"},"schema_version":"1.0","source":{"id":"1710.05996","kind":"arxiv","version":1}},"canonical_sha256":"89d3ab04920301b3928411fcbec7d1e0048f7ff6904af6ef6d59ea4f04f626dc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"89d3ab04920301b3928411fcbec7d1e0048f7ff6904af6ef6d59ea4f04f626dc","first_computed_at":"2026-05-18T00:32:38.074043Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:38.074043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BCoi68gNY6DpbVxRKd62v350l0PTswECFSBS+XS0sS2xJis/S2Bv5trcCKGlt0/GDDw1wv9o+ibjEAbMk6wPCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:38.074710Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.05996","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:669a2ac3411f578e9fdf035e5e3cb150b373403e65e06c91b673348b66da5aea","sha256:9d3d48b2e2c5ba8d4d12f279a2a5ad4e22eb800020941bbae2a3bd711bf99d93"],"state_sha256":"ffe71b8778d7965ef421a0017e5a2bd40d383446bdd0ffe2746b0c31ab68742a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dLZnetMJDEN368nq0VQEkRi/gTsjk2FuBL23A9KIEnlr6T3k1DuLeEjHSCAZ7yatArhZ1QV+r9s3h5jiChaUBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T14:43:08.861571Z","bundle_sha256":"6e8fb37f9df4e145f39a070463618001284b82405cb26e9e1b26bc84bb6690fc"}}