{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:WLSBIBV7N34YWZNFEDJWOATQOV","short_pith_number":"pith:WLSBIBV7","canonical_record":{"source":{"id":"1004.3321","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-04-19T22:52:25Z","cross_cats_sorted":[],"title_canon_sha256":"af435ab37de08259b6adea25161d33d6651588ed43d8395ca9aa2fc76a63d4d9","abstract_canon_sha256":"eebfda1dbcbd8815ec454c408b667cdb3fbc3f0f7304b4adaa2eab238a72200d"},"schema_version":"1.0"},"canonical_sha256":"b2e41406bf6ef98b65a520d36702707554cdf7792de00f5f4e2cd727949a481d","source":{"kind":"arxiv","id":"1004.3321","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.3321","created_at":"2026-05-18T04:01:13Z"},{"alias_kind":"arxiv_version","alias_value":"1004.3321v4","created_at":"2026-05-18T04:01:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.3321","created_at":"2026-05-18T04:01:13Z"},{"alias_kind":"pith_short_12","alias_value":"WLSBIBV7N34Y","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WLSBIBV7N34YWZNF","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WLSBIBV7","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:WLSBIBV7N34YWZNFEDJWOATQOV","target":"record","payload":{"canonical_record":{"source":{"id":"1004.3321","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-04-19T22:52:25Z","cross_cats_sorted":[],"title_canon_sha256":"af435ab37de08259b6adea25161d33d6651588ed43d8395ca9aa2fc76a63d4d9","abstract_canon_sha256":"eebfda1dbcbd8815ec454c408b667cdb3fbc3f0f7304b4adaa2eab238a72200d"},"schema_version":"1.0"},"canonical_sha256":"b2e41406bf6ef98b65a520d36702707554cdf7792de00f5f4e2cd727949a481d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:01:13.436587Z","signature_b64":"0Ky8mKTFyTTJgkxs9DWbvKT97ZZeabe4xcf3Pul39eZQnhmgXROuPs0Or3d5Nw0Mj/wyvowS17PqTfoOzdsdAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2e41406bf6ef98b65a520d36702707554cdf7792de00f5f4e2cd727949a481d","last_reissued_at":"2026-05-18T04:01:13.435830Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:01:13.435830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.3321","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:01:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aXy8Q3g/x3eQ1ITzGHd3bBmkhwVmudalTBxxNhZiZReZZJPY7KKXY/Jb0AIL6vFlCMwzzB/Q9BEk9Q5lsllrCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:03:43.159616Z"},"content_sha256":"ead0f8a7eae3b4ad6f0727a9ec115ae15c76feba9dc0a4973a3d8e895674d78c","schema_version":"1.0","event_id":"sha256:ead0f8a7eae3b4ad6f0727a9ec115ae15c76feba9dc0a4973a3d8e895674d78c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:WLSBIBV7N34YWZNFEDJWOATQOV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Sandpile group of the cone of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Carlos A. Alfaro, Carlos E. Valencia","submitted_at":"2010-04-19T22:52:25Z","abstract_excerpt":"In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. As an application of these result we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube of dimension d."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.3321","kind":"arxiv","version":4},"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":"2026-05-18T04:01:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VA2voZTphoFIZdzkRUWIog9ebuERmcoA36+UXm+pvDCPQmJol+1YLetP+3qqSo+21D54+JdJgIwOcMLqYWmfBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:03:43.160178Z"},"content_sha256":"a0bce289c2a63b70a0501a551054d005d3980ddf6358f77a03620e350c5215f8","schema_version":"1.0","event_id":"sha256:a0bce289c2a63b70a0501a551054d005d3980ddf6358f77a03620e350c5215f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WLSBIBV7N34YWZNFEDJWOATQOV/bundle.json","state_url":"https://pith.science/pith/WLSBIBV7N34YWZNFEDJWOATQOV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WLSBIBV7N34YWZNFEDJWOATQOV/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":"2026-06-07T11:03:43Z","links":{"resolver":"https://pith.science/pith/WLSBIBV7N34YWZNFEDJWOATQOV","bundle":"https://pith.science/pith/WLSBIBV7N34YWZNFEDJWOATQOV/bundle.json","state":"https://pith.science/pith/WLSBIBV7N34YWZNFEDJWOATQOV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WLSBIBV7N34YWZNFEDJWOATQOV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:WLSBIBV7N34YWZNFEDJWOATQOV","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":"eebfda1dbcbd8815ec454c408b667cdb3fbc3f0f7304b4adaa2eab238a72200d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-04-19T22:52:25Z","title_canon_sha256":"af435ab37de08259b6adea25161d33d6651588ed43d8395ca9aa2fc76a63d4d9"},"schema_version":"1.0","source":{"id":"1004.3321","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.3321","created_at":"2026-05-18T04:01:13Z"},{"alias_kind":"arxiv_version","alias_value":"1004.3321v4","created_at":"2026-05-18T04:01:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.3321","created_at":"2026-05-18T04:01:13Z"},{"alias_kind":"pith_short_12","alias_value":"WLSBIBV7N34Y","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"WLSBIBV7N34YWZNF","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"WLSBIBV7","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:a0bce289c2a63b70a0501a551054d005d3980ddf6358f77a03620e350c5215f8","target":"graph","created_at":"2026-05-18T04:01:13Z","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":"In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an injective homomorphism between their sandpile groups. As an application of these result we obtain an explicit description of a set of generators of the sandpile group of the cone of the hypercube of dimension d.","authors_text":"Carlos A. Alfaro, Carlos E. Valencia","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-04-19T22:52:25Z","title":"On the Sandpile group of the cone of a graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.3321","kind":"arxiv","version":4},"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:ead0f8a7eae3b4ad6f0727a9ec115ae15c76feba9dc0a4973a3d8e895674d78c","target":"record","created_at":"2026-05-18T04:01:13Z","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":"eebfda1dbcbd8815ec454c408b667cdb3fbc3f0f7304b4adaa2eab238a72200d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-04-19T22:52:25Z","title_canon_sha256":"af435ab37de08259b6adea25161d33d6651588ed43d8395ca9aa2fc76a63d4d9"},"schema_version":"1.0","source":{"id":"1004.3321","kind":"arxiv","version":4}},"canonical_sha256":"b2e41406bf6ef98b65a520d36702707554cdf7792de00f5f4e2cd727949a481d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b2e41406bf6ef98b65a520d36702707554cdf7792de00f5f4e2cd727949a481d","first_computed_at":"2026-05-18T04:01:13.435830Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:01:13.435830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0Ky8mKTFyTTJgkxs9DWbvKT97ZZeabe4xcf3Pul39eZQnhmgXROuPs0Or3d5Nw0Mj/wyvowS17PqTfoOzdsdAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:01:13.436587Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.3321","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ead0f8a7eae3b4ad6f0727a9ec115ae15c76feba9dc0a4973a3d8e895674d78c","sha256:a0bce289c2a63b70a0501a551054d005d3980ddf6358f77a03620e350c5215f8"],"state_sha256":"183885582e55e4f1955e3628b6c5256c156d99db04ebb25345c097b9c46fa061"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E8Tu7zHrao06+oitWqaeec9sZEZTyIEbpAb/sjmGkpUCSKE2xtmnSmwschULI0wM6OOQfCbI+l8MHKucL4RRCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T11:03:43.163475Z","bundle_sha256":"d616a2a8de114694a7f97b2f533d7505afe5a435c0eda2685ebdec2165a56560"}}