{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:PVJLEJLAAV4SFKYTKUDFDNW4O2","short_pith_number":"pith:PVJLEJLA","canonical_record":{"source":{"id":"1308.5340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-08-24T14:47:41Z","cross_cats_sorted":[],"title_canon_sha256":"e0073bc124f0a5c5f24a966995b593a9389afca99d0f84b5d7b15f377c36a54b","abstract_canon_sha256":"9f4fa1cd6619a5faee0c7fe463fdc0dd4cbe9a2fe1636823a07a2e10cb7ff921"},"schema_version":"1.0"},"canonical_sha256":"7d52b22560057922ab13550651b6dc769d4d9883e4dac34fa8bc60265f30f1fd","source":{"kind":"arxiv","id":"1308.5340","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5340","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5340v1","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5340","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"pith_short_12","alias_value":"PVJLEJLAAV4S","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"PVJLEJLAAV4SFKYT","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"PVJLEJLA","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:PVJLEJLAAV4SFKYTKUDFDNW4O2","target":"record","payload":{"canonical_record":{"source":{"id":"1308.5340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-08-24T14:47:41Z","cross_cats_sorted":[],"title_canon_sha256":"e0073bc124f0a5c5f24a966995b593a9389afca99d0f84b5d7b15f377c36a54b","abstract_canon_sha256":"9f4fa1cd6619a5faee0c7fe463fdc0dd4cbe9a2fe1636823a07a2e10cb7ff921"},"schema_version":"1.0"},"canonical_sha256":"7d52b22560057922ab13550651b6dc769d4d9883e4dac34fa8bc60265f30f1fd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:15:05.039243Z","signature_b64":"g1Ffal2KNiL0PQ8ovkHnRaY/tjsLM6hkgKdV37DEX31NXvFJ0gOKMiBRW0qtKrXxcB3PQgAIhWX4DDKkaHNQBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d52b22560057922ab13550651b6dc769d4d9883e4dac34fa8bc60265f30f1fd","last_reissued_at":"2026-05-18T03:15:05.038459Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:15:05.038459Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.5340","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":"2026-05-18T03:15:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KXlcicOq38RlAXRDRIKs7RDe/xk+xH6u0+hU/D0S3f8a37UMCx2b566D2s7KZ0lnHJuo8l421Ecaook0lSPUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:32:46.488211Z"},"content_sha256":"9edb8aee96a82fe5696950a3dbfcd6a2b0a46a032491f7e9e5d78fd7f277bfec","schema_version":"1.0","event_id":"sha256:9edb8aee96a82fe5696950a3dbfcd6a2b0a46a032491f7e9e5d78fd7f277bfec"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:PVJLEJLAAV4SFKYTKUDFDNW4O2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On sums of graph eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Evans M. Harell II, Joachim Stubbe","submitted_at":"2013-08-24T14:47:41Z","abstract_excerpt":"We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both the standard combinatorial Laplacian and the renormalized Laplacian. We also provide upper bounds for sums of squares of eigenvalues of these three matrices.\n  Among our results, we generalize an inequality of Fiedler for the extreme eigenvalues of the graph Laplacian to a bound on the sums of the smallest (or largest) k such eigenvalues, k < n.\n  Furthermore,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5340","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":"2026-05-18T03:15:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XdHMwupThU0Iyia0J5MQsxzNfJ09Iv80W+zGqc6S1MKPHH2T926nNTFS9Ic1U/SvB/E9yjXQx1bhJfoMLVyVBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:32:46.488925Z"},"content_sha256":"1b3cf756cab3e972eee95ea09adb0cc82210c211562350979bcd994882c33b63","schema_version":"1.0","event_id":"sha256:1b3cf756cab3e972eee95ea09adb0cc82210c211562350979bcd994882c33b63"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PVJLEJLAAV4SFKYTKUDFDNW4O2/bundle.json","state_url":"https://pith.science/pith/PVJLEJLAAV4SFKYTKUDFDNW4O2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PVJLEJLAAV4SFKYTKUDFDNW4O2/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":"2026-05-27T14:32:46Z","links":{"resolver":"https://pith.science/pith/PVJLEJLAAV4SFKYTKUDFDNW4O2","bundle":"https://pith.science/pith/PVJLEJLAAV4SFKYTKUDFDNW4O2/bundle.json","state":"https://pith.science/pith/PVJLEJLAAV4SFKYTKUDFDNW4O2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PVJLEJLAAV4SFKYTKUDFDNW4O2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:PVJLEJLAAV4SFKYTKUDFDNW4O2","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":"9f4fa1cd6619a5faee0c7fe463fdc0dd4cbe9a2fe1636823a07a2e10cb7ff921","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-08-24T14:47:41Z","title_canon_sha256":"e0073bc124f0a5c5f24a966995b593a9389afca99d0f84b5d7b15f377c36a54b"},"schema_version":"1.0","source":{"id":"1308.5340","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5340","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5340v1","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5340","created_at":"2026-05-18T03:15:05Z"},{"alias_kind":"pith_short_12","alias_value":"PVJLEJLAAV4S","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"PVJLEJLAAV4SFKYT","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"PVJLEJLA","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:1b3cf756cab3e972eee95ea09adb0cc82210c211562350979bcd994882c33b63","target":"graph","created_at":"2026-05-18T03:15:05Z","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 use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both the standard combinatorial Laplacian and the renormalized Laplacian. We also provide upper bounds for sums of squares of eigenvalues of these three matrices.\n  Among our results, we generalize an inequality of Fiedler for the extreme eigenvalues of the graph Laplacian to a bound on the sums of the smallest (or largest) k such eigenvalues, k < n.\n  Furthermore,","authors_text":"Evans M. Harell II, Joachim Stubbe","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-08-24T14:47:41Z","title":"On sums of graph eigenvalues"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5340","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":"sha256:9edb8aee96a82fe5696950a3dbfcd6a2b0a46a032491f7e9e5d78fd7f277bfec","target":"record","created_at":"2026-05-18T03:15:05Z","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":"9f4fa1cd6619a5faee0c7fe463fdc0dd4cbe9a2fe1636823a07a2e10cb7ff921","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-08-24T14:47:41Z","title_canon_sha256":"e0073bc124f0a5c5f24a966995b593a9389afca99d0f84b5d7b15f377c36a54b"},"schema_version":"1.0","source":{"id":"1308.5340","kind":"arxiv","version":1}},"canonical_sha256":"7d52b22560057922ab13550651b6dc769d4d9883e4dac34fa8bc60265f30f1fd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d52b22560057922ab13550651b6dc769d4d9883e4dac34fa8bc60265f30f1fd","first_computed_at":"2026-05-18T03:15:05.038459Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:05.038459Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g1Ffal2KNiL0PQ8ovkHnRaY/tjsLM6hkgKdV37DEX31NXvFJ0gOKMiBRW0qtKrXxcB3PQgAIhWX4DDKkaHNQBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:05.039243Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.5340","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9edb8aee96a82fe5696950a3dbfcd6a2b0a46a032491f7e9e5d78fd7f277bfec","sha256:1b3cf756cab3e972eee95ea09adb0cc82210c211562350979bcd994882c33b63"],"state_sha256":"16aaf9240c4e87956ea9685eb2c7d1ef079c7cbb0cfced635119f38fda992211"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2+7o1VGYQWkKnyf4wVYRm/4v7Vl+OCEqarD3Xi7HuIxT4wP3iLneICXJPHlKTaMDsFcjeggL06X2NtytWhtjBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:32:46.492780Z","bundle_sha256":"6ec1e591717f1a99a3edb303aac65bf24820b2812f264d84bad1da197fbcfe5d"}}