{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:BHRXHMAPWPG6CBPUBGW2HKP3QL","short_pith_number":"pith:BHRXHMAP","canonical_record":{"source":{"id":"1709.04009","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-12T18:29:44Z","cross_cats_sorted":[],"title_canon_sha256":"1b28242535783036838c07ca36a40071e1ae8fd46c807f55eead613acc28422f","abstract_canon_sha256":"e3e742ae1eb6bbbb4c1e428b74b33083587d8260c8f007a137f41200a6d3d1ef"},"schema_version":"1.0"},"canonical_sha256":"09e373b00fb3cde105f409ada3a9fb82d9445a390ce99868d520ed3eb36008ed","source":{"kind":"arxiv","id":"1709.04009","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.04009","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"arxiv_version","alias_value":"1709.04009v4","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.04009","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"pith_short_12","alias_value":"BHRXHMAPWPG6","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BHRXHMAPWPG6CBPU","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BHRXHMAP","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:BHRXHMAPWPG6CBPUBGW2HKP3QL","target":"record","payload":{"canonical_record":{"source":{"id":"1709.04009","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-12T18:29:44Z","cross_cats_sorted":[],"title_canon_sha256":"1b28242535783036838c07ca36a40071e1ae8fd46c807f55eead613acc28422f","abstract_canon_sha256":"e3e742ae1eb6bbbb4c1e428b74b33083587d8260c8f007a137f41200a6d3d1ef"},"schema_version":"1.0"},"canonical_sha256":"09e373b00fb3cde105f409ada3a9fb82d9445a390ce99868d520ed3eb36008ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:18.912608Z","signature_b64":"2ewBwSBCGpK2ONOMyRNTk+/pSB0Ir0vTeTWKIayr5dUQSPqebYaawicOTnhk0iMQRte6KLwBS07D4bzY59WjDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09e373b00fb3cde105f409ada3a9fb82d9445a390ce99868d520ed3eb36008ed","last_reissued_at":"2026-05-17T23:52:18.911847Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:18.911847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.04009","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":"2026-05-17T23:52:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TTi2kp86vg/J7/hOnsqZ8pgN0P3245Caz4gRgGqX+4+m+QhScf5ubilx3ejh0VeO5ja27DZNH/0gtJAYkNUuDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:47:16.658300Z"},"content_sha256":"3951c9b166baa8c7eac57775a3d57f48e8c47a7fecbf2a609fa4c5d025abcd26","schema_version":"1.0","event_id":"sha256:3951c9b166baa8c7eac57775a3d57f48e8c47a7fecbf2a609fa4c5d025abcd26"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:BHRXHMAPWPG6CBPUBGW2HKP3QL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Conjectured bound for the distribution of eigenvalues of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Clive Elphick, Pawel Wocjan","submitted_at":"2017-09-12T18:29:44Z","abstract_excerpt":"Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \\[ n^-(G) + n^-(\\bar{G}) \\le 1.5(n - 1), \\] and prove this bound for various classes of graphs and for almost all graphs.\n  We consider the relationship between this bound and the number of eigenvalues that lie within the interval $-1$ to $0$, which we denote $n_{(-1,0)}(G)$. We conjecture that for any graph \\[ n_{(-1,0)}(G) \\le 0.5(n - 1). \\] and prove this bound for almost all graphs. We also investigate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04009","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":"2026-05-17T23:52:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"urO8eBPqLfbvp3ufx9ojGNfmJWI185CBo34q6ncwa3xbcnrpv/xVRhx5YJNBL9uFLdRBTfUwVDHYlZ3jk7BqCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:47:16.658697Z"},"content_sha256":"b54280adde817aca7837fd4a1109fa62ce16b34b02f350f4c58b9ced10b90b2f","schema_version":"1.0","event_id":"sha256:b54280adde817aca7837fd4a1109fa62ce16b34b02f350f4c58b9ced10b90b2f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BHRXHMAPWPG6CBPUBGW2HKP3QL/bundle.json","state_url":"https://pith.science/pith/BHRXHMAPWPG6CBPUBGW2HKP3QL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BHRXHMAPWPG6CBPUBGW2HKP3QL/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":"2026-05-27T15:47:16Z","links":{"resolver":"https://pith.science/pith/BHRXHMAPWPG6CBPUBGW2HKP3QL","bundle":"https://pith.science/pith/BHRXHMAPWPG6CBPUBGW2HKP3QL/bundle.json","state":"https://pith.science/pith/BHRXHMAPWPG6CBPUBGW2HKP3QL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BHRXHMAPWPG6CBPUBGW2HKP3QL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BHRXHMAPWPG6CBPUBGW2HKP3QL","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":"e3e742ae1eb6bbbb4c1e428b74b33083587d8260c8f007a137f41200a6d3d1ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-12T18:29:44Z","title_canon_sha256":"1b28242535783036838c07ca36a40071e1ae8fd46c807f55eead613acc28422f"},"schema_version":"1.0","source":{"id":"1709.04009","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.04009","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"arxiv_version","alias_value":"1709.04009v4","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.04009","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"pith_short_12","alias_value":"BHRXHMAPWPG6","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BHRXHMAPWPG6CBPU","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BHRXHMAP","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:b54280adde817aca7837fd4a1109fa62ce16b34b02f350f4c58b9ced10b90b2f","target":"graph","created_at":"2026-05-17T23:52:18Z","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 $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \\[ n^-(G) + n^-(\\bar{G}) \\le 1.5(n - 1), \\] and prove this bound for various classes of graphs and for almost all graphs.\n  We consider the relationship between this bound and the number of eigenvalues that lie within the interval $-1$ to $0$, which we denote $n_{(-1,0)}(G)$. We conjecture that for any graph \\[ n_{(-1,0)}(G) \\le 0.5(n - 1). \\] and prove this bound for almost all graphs. We also investigate","authors_text":"Clive Elphick, Pawel Wocjan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-12T18:29:44Z","title":"Conjectured bound for the distribution of eigenvalues of a graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04009","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":"sha256:3951c9b166baa8c7eac57775a3d57f48e8c47a7fecbf2a609fa4c5d025abcd26","target":"record","created_at":"2026-05-17T23:52:18Z","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":"e3e742ae1eb6bbbb4c1e428b74b33083587d8260c8f007a137f41200a6d3d1ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-12T18:29:44Z","title_canon_sha256":"1b28242535783036838c07ca36a40071e1ae8fd46c807f55eead613acc28422f"},"schema_version":"1.0","source":{"id":"1709.04009","kind":"arxiv","version":4}},"canonical_sha256":"09e373b00fb3cde105f409ada3a9fb82d9445a390ce99868d520ed3eb36008ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09e373b00fb3cde105f409ada3a9fb82d9445a390ce99868d520ed3eb36008ed","first_computed_at":"2026-05-17T23:52:18.911847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:18.911847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2ewBwSBCGpK2ONOMyRNTk+/pSB0Ir0vTeTWKIayr5dUQSPqebYaawicOTnhk0iMQRte6KLwBS07D4bzY59WjDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:18.912608Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.04009","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3951c9b166baa8c7eac57775a3d57f48e8c47a7fecbf2a609fa4c5d025abcd26","sha256:b54280adde817aca7837fd4a1109fa62ce16b34b02f350f4c58b9ced10b90b2f"],"state_sha256":"4d9c45f5d7979551eface4f1665b80ec7625ae6b60c412d9ad1cba34db266ed8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XIGn4QI9naDdlNviC132bWJle6ECL7UjtPYvU4gU9ni8Ua65TSuE+R1B8dKK1HHuph8eG5CpEx4qzK9638PPCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T15:47:16.661003Z","bundle_sha256":"dde0631ef2c3eb204db27d31dea7601902e301b9af56cbe22c337fed97be1d5f"}}