{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:VFH3XE7O57UCABN24YHARYO3DT","short_pith_number":"pith:VFH3XE7O","canonical_record":{"source":{"id":"1304.1205","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T22:37:38Z","cross_cats_sorted":[],"title_canon_sha256":"52467f4a440a4d9d7ed6cac8ac8d9265ba93f61cf45158f396a6cec2d209a991","abstract_canon_sha256":"aa0141f27d30ad39d64b3bbf2c2ea78c3ca7eafe96ab43d7b825daeedabd44a0"},"schema_version":"1.0"},"canonical_sha256":"a94fbb93eeefe82005bae60e08e1db1cc60a92269cc2f6dc1b6b1031d8c0bab7","source":{"kind":"arxiv","id":"1304.1205","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.1205","created_at":"2026-05-18T03:29:04Z"},{"alias_kind":"arxiv_version","alias_value":"1304.1205v1","created_at":"2026-05-18T03:29:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1205","created_at":"2026-05-18T03:29:04Z"},{"alias_kind":"pith_short_12","alias_value":"VFH3XE7O57UC","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VFH3XE7O57UCABN2","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VFH3XE7O","created_at":"2026-05-18T12:28:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:VFH3XE7O57UCABN24YHARYO3DT","target":"record","payload":{"canonical_record":{"source":{"id":"1304.1205","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T22:37:38Z","cross_cats_sorted":[],"title_canon_sha256":"52467f4a440a4d9d7ed6cac8ac8d9265ba93f61cf45158f396a6cec2d209a991","abstract_canon_sha256":"aa0141f27d30ad39d64b3bbf2c2ea78c3ca7eafe96ab43d7b825daeedabd44a0"},"schema_version":"1.0"},"canonical_sha256":"a94fbb93eeefe82005bae60e08e1db1cc60a92269cc2f6dc1b6b1031d8c0bab7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:04.259952Z","signature_b64":"pnPaNBaWAqItWteskDMp1fNmwGvqqQ7l8GtnksMzCzbojVlKbPtBXsAYHAUh2VJb16B8xNudKdV0x8QFfAKfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a94fbb93eeefe82005bae60e08e1db1cc60a92269cc2f6dc1b6b1031d8c0bab7","last_reissued_at":"2026-05-18T03:29:04.259260Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:04.259260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.1205","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":"2026-05-18T03:29:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"10VrjRU57+n5NHR9Up68DYfuBoCWPZohwutSL5mXumSfvwD88IrlglqXNpfqOwiZ6mLKK5x1ZHSZeWa0SFOXCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T14:38:43.281604Z"},"content_sha256":"34ff0e61febf8fc26e16f21cb275f885c60d5744a746d21d041d73724601454b","schema_version":"1.0","event_id":"sha256:34ff0e61febf8fc26e16f21cb275f885c60d5744a746d21d041d73724601454b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:VFH3XE7O57UCABN24YHARYO3DT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimum number of distinct eigenvalues of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bahman Ahmadi, Fatemeh Alinaghipour, Karen Meagher, Michael S. Cavers, Shahla Nasserasr, Shaun Fallat","submitted_at":"2013-04-03T22:37:38Z","abstract_excerpt":"The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce further properties of $q(G)$. It is shown that there is a great number of graphs $G$ for which $q(G)=2$. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs $G$ are provided to show that adding and deleting edges or vertices can dramatically chang"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1205","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":"2026-05-18T03:29:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZGLZwNUGLxicpxemk6JxkMR+1ynkOZlgfV7QqrVr0A1ZrigAzdk2m8bsld6bR0LkRk5FMG6WsKhIJoXry8M1Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T14:38:43.281964Z"},"content_sha256":"56531ef50d499fea5987e497f0b7bfd04ab51e7197a8e681f82d05d6aaa03640","schema_version":"1.0","event_id":"sha256:56531ef50d499fea5987e497f0b7bfd04ab51e7197a8e681f82d05d6aaa03640"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VFH3XE7O57UCABN24YHARYO3DT/bundle.json","state_url":"https://pith.science/pith/VFH3XE7O57UCABN24YHARYO3DT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VFH3XE7O57UCABN24YHARYO3DT/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":"2026-05-28T14:38:43Z","links":{"resolver":"https://pith.science/pith/VFH3XE7O57UCABN24YHARYO3DT","bundle":"https://pith.science/pith/VFH3XE7O57UCABN24YHARYO3DT/bundle.json","state":"https://pith.science/pith/VFH3XE7O57UCABN24YHARYO3DT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VFH3XE7O57UCABN24YHARYO3DT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:VFH3XE7O57UCABN24YHARYO3DT","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":"aa0141f27d30ad39d64b3bbf2c2ea78c3ca7eafe96ab43d7b825daeedabd44a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T22:37:38Z","title_canon_sha256":"52467f4a440a4d9d7ed6cac8ac8d9265ba93f61cf45158f396a6cec2d209a991"},"schema_version":"1.0","source":{"id":"1304.1205","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.1205","created_at":"2026-05-18T03:29:04Z"},{"alias_kind":"arxiv_version","alias_value":"1304.1205v1","created_at":"2026-05-18T03:29:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.1205","created_at":"2026-05-18T03:29:04Z"},{"alias_kind":"pith_short_12","alias_value":"VFH3XE7O57UC","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VFH3XE7O57UCABN2","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VFH3XE7O","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:56531ef50d499fea5987e497f0b7bfd04ab51e7197a8e681f82d05d6aaa03640","target":"graph","created_at":"2026-05-18T03:29:04Z","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 minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce further properties of $q(G)$. It is shown that there is a great number of graphs $G$ for which $q(G)=2$. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs $G$ are provided to show that adding and deleting edges or vertices can dramatically chang","authors_text":"Bahman Ahmadi, Fatemeh Alinaghipour, Karen Meagher, Michael S. Cavers, Shahla Nasserasr, Shaun Fallat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T22:37:38Z","title":"Minimum number of distinct eigenvalues of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1205","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":"sha256:34ff0e61febf8fc26e16f21cb275f885c60d5744a746d21d041d73724601454b","target":"record","created_at":"2026-05-18T03:29:04Z","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":"aa0141f27d30ad39d64b3bbf2c2ea78c3ca7eafe96ab43d7b825daeedabd44a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-03T22:37:38Z","title_canon_sha256":"52467f4a440a4d9d7ed6cac8ac8d9265ba93f61cf45158f396a6cec2d209a991"},"schema_version":"1.0","source":{"id":"1304.1205","kind":"arxiv","version":1}},"canonical_sha256":"a94fbb93eeefe82005bae60e08e1db1cc60a92269cc2f6dc1b6b1031d8c0bab7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a94fbb93eeefe82005bae60e08e1db1cc60a92269cc2f6dc1b6b1031d8c0bab7","first_computed_at":"2026-05-18T03:29:04.259260Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:04.259260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pnPaNBaWAqItWteskDMp1fNmwGvqqQ7l8GtnksMzCzbojVlKbPtBXsAYHAUh2VJb16B8xNudKdV0x8QFfAKfAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:04.259952Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.1205","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34ff0e61febf8fc26e16f21cb275f885c60d5744a746d21d041d73724601454b","sha256:56531ef50d499fea5987e497f0b7bfd04ab51e7197a8e681f82d05d6aaa03640"],"state_sha256":"28ad4e71cb08a1842de724aaa153acbf054a54da6e8a9f28fc1a427bc804ee6e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4RI2mhgMrcHjkmDHEaaZExMq4JXhkNlLaetsq5IQVsjo4Qt0k4Y5E5vVrK2xpcM8XCi8MJ32JC1vTdb4q6rmAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T14:38:43.283962Z","bundle_sha256":"1c17c4f18c6e58611e127015f96a818bb73b592ca8fb35f63402d22d6b9a90a3"}}