{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:KITBKAU55QWDUP3IBZHQDTJYEC","short_pith_number":"pith:KITBKAU5","canonical_record":{"source":{"id":"1209.3190","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-14T13:42:53Z","cross_cats_sorted":["cs.DM","quant-ph"],"title_canon_sha256":"cd277e6656aa7fcdf02486d8d5800c88b29ff05569992635bfcfd08b46610114","abstract_canon_sha256":"21c8461a94c9cda7bb72d5a6e28aee7e45e33ac9987c1fdb6db2735972266522"},"schema_version":"1.0"},"canonical_sha256":"522615029dec2c3a3f680e4f01cd3820b20898da7f1fc7828d1fe6a7db0ec054","source":{"kind":"arxiv","id":"1209.3190","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3190","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3190v1","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3190","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"pith_short_12","alias_value":"KITBKAU55QWD","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"KITBKAU55QWDUP3I","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"KITBKAU5","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:KITBKAU55QWDUP3IBZHQDTJYEC","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3190","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-14T13:42:53Z","cross_cats_sorted":["cs.DM","quant-ph"],"title_canon_sha256":"cd277e6656aa7fcdf02486d8d5800c88b29ff05569992635bfcfd08b46610114","abstract_canon_sha256":"21c8461a94c9cda7bb72d5a6e28aee7e45e33ac9987c1fdb6db2735972266522"},"schema_version":"1.0"},"canonical_sha256":"522615029dec2c3a3f680e4f01cd3820b20898da7f1fc7828d1fe6a7db0ec054","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:32.567024Z","signature_b64":"paHhu+TtH3hN+X6pH3yvjtY6kkEYdWhpl/3QqnUxATFvU96DhnDtrirWgrCZM+k7YN2Kr105fagtjGCfiaimBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"522615029dec2c3a3f680e4f01cd3820b20898da7f1fc7828d1fe6a7db0ec054","last_reissued_at":"2026-05-18T03:45:32.566350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:32.566350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3190","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":"2026-05-18T03:45:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oMGxuXAMYrquMn2LLuO+jnilQrRUStBDTCdJ3u0fXuFixtlTtRDAuAYPuTbCyutrjjTlmz+guAY6RzkUObDbBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:15:42.639916Z"},"content_sha256":"605047246afa30009cc5cf717ec12f6e19f54dd770dc168201d38e620a44b548","schema_version":"1.0","event_id":"sha256:605047246afa30009cc5cf717ec12f6e19f54dd770dc168201d38e620a44b548"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:KITBKAU55QWDUP3IBZHQDTJYEC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","quant-ph"],"primary_cat":"math.CO","authors_text":"Clive Elphick, Pawel Wocjan","submitted_at":"2012-09-14T13:42:53Z","abstract_excerpt":"The purpose of this article is to improve existing lower bounds on the chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency matrix sorted in non-increasing order.\n  First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / - sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the {\\sc Hoffman} lower bound.\n  Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^- are the s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3190","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":"2026-05-18T03:45:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xq/rVa5Q/44u6NumD7jKBlescp7iAoOFPyvVemD1o+4vIsaz7RO+p5srnHyld6dXnE2WtyptMvb2Ql1maMDsBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:15:42.640261Z"},"content_sha256":"1f0135ca82c0a6801c48db03a1e79c9ce13aa9983b62c5e8ae0d9e5890f11aff","schema_version":"1.0","event_id":"sha256:1f0135ca82c0a6801c48db03a1e79c9ce13aa9983b62c5e8ae0d9e5890f11aff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KITBKAU55QWDUP3IBZHQDTJYEC/bundle.json","state_url":"https://pith.science/pith/KITBKAU55QWDUP3IBZHQDTJYEC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KITBKAU55QWDUP3IBZHQDTJYEC/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":"2026-05-28T06:15:42Z","links":{"resolver":"https://pith.science/pith/KITBKAU55QWDUP3IBZHQDTJYEC","bundle":"https://pith.science/pith/KITBKAU55QWDUP3IBZHQDTJYEC/bundle.json","state":"https://pith.science/pith/KITBKAU55QWDUP3IBZHQDTJYEC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KITBKAU55QWDUP3IBZHQDTJYEC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:KITBKAU55QWDUP3IBZHQDTJYEC","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":"21c8461a94c9cda7bb72d5a6e28aee7e45e33ac9987c1fdb6db2735972266522","cross_cats_sorted":["cs.DM","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-14T13:42:53Z","title_canon_sha256":"cd277e6656aa7fcdf02486d8d5800c88b29ff05569992635bfcfd08b46610114"},"schema_version":"1.0","source":{"id":"1209.3190","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3190","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3190v1","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3190","created_at":"2026-05-18T03:45:32Z"},{"alias_kind":"pith_short_12","alias_value":"KITBKAU55QWD","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"KITBKAU55QWDUP3I","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"KITBKAU5","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:1f0135ca82c0a6801c48db03a1e79c9ce13aa9983b62c5e8ae0d9e5890f11aff","target":"graph","created_at":"2026-05-18T03:45:32Z","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":"The purpose of this article is to improve existing lower bounds on the chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency matrix sorted in non-increasing order.\n  First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / - sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the {\\sc Hoffman} lower bound.\n  Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^- are the s","authors_text":"Clive Elphick, Pawel Wocjan","cross_cats":["cs.DM","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-14T13:42:53Z","title":"New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3190","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":"sha256:605047246afa30009cc5cf717ec12f6e19f54dd770dc168201d38e620a44b548","target":"record","created_at":"2026-05-18T03:45:32Z","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":"21c8461a94c9cda7bb72d5a6e28aee7e45e33ac9987c1fdb6db2735972266522","cross_cats_sorted":["cs.DM","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-14T13:42:53Z","title_canon_sha256":"cd277e6656aa7fcdf02486d8d5800c88b29ff05569992635bfcfd08b46610114"},"schema_version":"1.0","source":{"id":"1209.3190","kind":"arxiv","version":1}},"canonical_sha256":"522615029dec2c3a3f680e4f01cd3820b20898da7f1fc7828d1fe6a7db0ec054","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"522615029dec2c3a3f680e4f01cd3820b20898da7f1fc7828d1fe6a7db0ec054","first_computed_at":"2026-05-18T03:45:32.566350Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:32.566350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"paHhu+TtH3hN+X6pH3yvjtY6kkEYdWhpl/3QqnUxATFvU96DhnDtrirWgrCZM+k7YN2Kr105fagtjGCfiaimBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:32.567024Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3190","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:605047246afa30009cc5cf717ec12f6e19f54dd770dc168201d38e620a44b548","sha256:1f0135ca82c0a6801c48db03a1e79c9ce13aa9983b62c5e8ae0d9e5890f11aff"],"state_sha256":"c3cff8fa2b0928cf185dcee38489c9edf8dec0b0d3f2fc4d60d159fbb52ee09d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mm1I9vB5Bg0wBlxTBfKbgN7GYYV0KaxZeGVuJlyHZ7W8o22xmIMk8kVKM7UeHwXamGgZtVkT+dSD79PAT0xrAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T06:15:42.642164Z","bundle_sha256":"545f12ac00de2f90627681e786ab48838708517588802e516900ced27d8e2401"}}