{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:2GFUZ2I6QPDA6XOTWIVPLL25ZV","short_pith_number":"pith:2GFUZ2I6","canonical_record":{"source":{"id":"1808.03932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-12T12:35:16Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"3e00674a0b79056bdf85a8af5834be78fa2a250ac61c33436b38d3a407376d1b","abstract_canon_sha256":"e6a3ced63beecc4e72cfb9b519066798702d3914bf1162070e53c4b465628407"},"schema_version":"1.0"},"canonical_sha256":"d18b4ce91e83c60f5dd3b22af5af5dcd62f88057a410f0afca0023168e2e06dd","source":{"kind":"arxiv","id":"1808.03932","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.03932","created_at":"2026-05-18T00:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"1808.03932v1","created_at":"2026-05-18T00:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.03932","created_at":"2026-05-18T00:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"2GFUZ2I6QPDA","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2GFUZ2I6QPDA6XOT","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2GFUZ2I6","created_at":"2026-05-18T12:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:2GFUZ2I6QPDA6XOTWIVPLL25ZV","target":"record","payload":{"canonical_record":{"source":{"id":"1808.03932","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-12T12:35:16Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"3e00674a0b79056bdf85a8af5834be78fa2a250ac61c33436b38d3a407376d1b","abstract_canon_sha256":"e6a3ced63beecc4e72cfb9b519066798702d3914bf1162070e53c4b465628407"},"schema_version":"1.0"},"canonical_sha256":"d18b4ce91e83c60f5dd3b22af5af5dcd62f88057a410f0afca0023168e2e06dd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:14.207555Z","signature_b64":"u7AGx0SG2vUoT9lOEUmcAnRKNFYLjLKz06TGwi02BRgeFzq1Eu1C2VlfGWqit/LGgX/exgBj+wSX/iT5t3InAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d18b4ce91e83c60f5dd3b22af5af5dcd62f88057a410f0afca0023168e2e06dd","last_reissued_at":"2026-05-18T00:08:14.207184Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:14.207184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1808.03932","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":"2026-05-18T00:08:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9krsr30ijWTTOg9DPY0CNm58KIOK8jXmdymVsNdWC1cSsvT2dM3I07LO2WlnzFThKcXS82PtIBOvzpAlZEwbAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T19:03:54.723710Z"},"content_sha256":"5e4d2f651a24f4a47f9bee31473798af8feebac9fdf949385d6eac308df9fc00","schema_version":"1.0","event_id":"sha256:5e4d2f651a24f4a47f9bee31473798af8feebac9fdf949385d6eac308df9fc00"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:2GFUZ2I6QPDA6XOTWIVPLL25ZV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$PC$-polynomial of graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CO","authors_text":"Vsevolod Gubarev","submitted_at":"2018-08-12T12:35:16Z","abstract_excerpt":"We define $PC$-polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root $\\beta(G)$ of $PC$-polynomial of the corresponding graph.\n  The random algebra is defined in such way that its growth rate equals the largest root of $PC$-polynomial of random graph. We prove that for almost all graphs all sufficiently large real roots of $PC$-polynomial lie in neighbourhoods of roots of $PC$-polynomial of random graph. We show how to calculate the series expansions of the latter roots. The average "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03932","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":"2026-05-18T00:08:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VdvcJf/ZKTojC9bfLNuUvQo2P+UoJy/FlHIYN1WF5wtcw3VAwfEFiYoU4cfs645G0gYZXWa7luq5Cl06q677AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T19:03:54.724071Z"},"content_sha256":"6598ed088d68c01db8fbfeed58a74f5ccda9db93b6a1b6a11feba199c209dfb8","schema_version":"1.0","event_id":"sha256:6598ed088d68c01db8fbfeed58a74f5ccda9db93b6a1b6a11feba199c209dfb8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2GFUZ2I6QPDA6XOTWIVPLL25ZV/bundle.json","state_url":"https://pith.science/pith/2GFUZ2I6QPDA6XOTWIVPLL25ZV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2GFUZ2I6QPDA6XOTWIVPLL25ZV/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":"2026-06-10T19:03:54Z","links":{"resolver":"https://pith.science/pith/2GFUZ2I6QPDA6XOTWIVPLL25ZV","bundle":"https://pith.science/pith/2GFUZ2I6QPDA6XOTWIVPLL25ZV/bundle.json","state":"https://pith.science/pith/2GFUZ2I6QPDA6XOTWIVPLL25ZV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2GFUZ2I6QPDA6XOTWIVPLL25ZV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:2GFUZ2I6QPDA6XOTWIVPLL25ZV","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":"e6a3ced63beecc4e72cfb9b519066798702d3914bf1162070e53c4b465628407","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-12T12:35:16Z","title_canon_sha256":"3e00674a0b79056bdf85a8af5834be78fa2a250ac61c33436b38d3a407376d1b"},"schema_version":"1.0","source":{"id":"1808.03932","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.03932","created_at":"2026-05-18T00:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"1808.03932v1","created_at":"2026-05-18T00:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.03932","created_at":"2026-05-18T00:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"2GFUZ2I6QPDA","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"2GFUZ2I6QPDA6XOT","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"2GFUZ2I6","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:6598ed088d68c01db8fbfeed58a74f5ccda9db93b6a1b6a11feba199c209dfb8","target":"graph","created_at":"2026-05-18T00:08:14Z","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":"We define $PC$-polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root $\\beta(G)$ of $PC$-polynomial of the corresponding graph.\n  The random algebra is defined in such way that its growth rate equals the largest root of $PC$-polynomial of random graph. We prove that for almost all graphs all sufficiently large real roots of $PC$-polynomial lie in neighbourhoods of roots of $PC$-polynomial of random graph. We show how to calculate the series expansions of the latter roots. The average ","authors_text":"Vsevolod Gubarev","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-12T12:35:16Z","title":"$PC$-polynomial of graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03932","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":"sha256:5e4d2f651a24f4a47f9bee31473798af8feebac9fdf949385d6eac308df9fc00","target":"record","created_at":"2026-05-18T00:08:14Z","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":"e6a3ced63beecc4e72cfb9b519066798702d3914bf1162070e53c4b465628407","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-12T12:35:16Z","title_canon_sha256":"3e00674a0b79056bdf85a8af5834be78fa2a250ac61c33436b38d3a407376d1b"},"schema_version":"1.0","source":{"id":"1808.03932","kind":"arxiv","version":1}},"canonical_sha256":"d18b4ce91e83c60f5dd3b22af5af5dcd62f88057a410f0afca0023168e2e06dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d18b4ce91e83c60f5dd3b22af5af5dcd62f88057a410f0afca0023168e2e06dd","first_computed_at":"2026-05-18T00:08:14.207184Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:14.207184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u7AGx0SG2vUoT9lOEUmcAnRKNFYLjLKz06TGwi02BRgeFzq1Eu1C2VlfGWqit/LGgX/exgBj+wSX/iT5t3InAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:14.207555Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.03932","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e4d2f651a24f4a47f9bee31473798af8feebac9fdf949385d6eac308df9fc00","sha256:6598ed088d68c01db8fbfeed58a74f5ccda9db93b6a1b6a11feba199c209dfb8"],"state_sha256":"f915e6aa16ae03d4d56b5e970b9aec2cf36a9f70fd6e9f54b0d25232e45f8997"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZXvbzyIp/MStXXMxdNNIdDskgwErSHEjWA7kIIOC8Rc0d4WchS4rrBrOrBUAugKtgAAi+6Mm7by49pro9eUiDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T19:03:54.726303Z","bundle_sha256":"399f70c814aca3ca0bb43d3c9da8a628e92d5c2a59cedc7a5fe3e80f39dc4689"}}