{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:NFWAFLYRK7EPQDHROHN5WGIOT6","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":"c20c0644dd411a740ba0547155b8b8a5bdd81243057065114491f11c58616521","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-20T18:26:18Z","title_canon_sha256":"d8e665e3c1d0d26dbc9ada7dbad038f60129aaf45b302a021f8647d160dc26fd"},"schema_version":"1.0","source":{"id":"1010.4269","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4269","created_at":"2026-05-18T03:53:17Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4269v5","created_at":"2026-05-18T03:53:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4269","created_at":"2026-05-18T03:53:17Z"},{"alias_kind":"pith_short_12","alias_value":"NFWAFLYRK7EP","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"NFWAFLYRK7EPQDHR","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"NFWAFLYR","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:b0efaa5c416cf5aef433749445d6b5d372c2a79e8bfbebb63003284198ce2acb","target":"graph","created_at":"2026-05-18T03:53:17Z","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 show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers.\n  In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.","authors_text":"Hao Chen, J\\\"urgen Jost","cross_cats":["math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-20T18:26:18Z","title":"Minimum vertex covers and the spectrum of the normalized Laplacian on trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4269","kind":"arxiv","version":5},"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":"sha256:6bc00d4a7f54f05f0e907480527a56a80036b9e244966d21f20c7ce4da98fa03","target":"record","created_at":"2026-05-18T03:53:17Z","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":"c20c0644dd411a740ba0547155b8b8a5bdd81243057065114491f11c58616521","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-10-20T18:26:18Z","title_canon_sha256":"d8e665e3c1d0d26dbc9ada7dbad038f60129aaf45b302a021f8647d160dc26fd"},"schema_version":"1.0","source":{"id":"1010.4269","kind":"arxiv","version":5}},"canonical_sha256":"696c02af1157c8f80cf171dbdb190e9f8217c20ec941c8fc5b5de13ecf7e71e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"696c02af1157c8f80cf171dbdb190e9f8217c20ec941c8fc5b5de13ecf7e71e2","first_computed_at":"2026-05-18T03:53:17.712133Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:17.712133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b/tvYMcQSNhoBs5ewNPrHwgpKLEhzWlEixZzeinfT1sYQ16SUTYXT9WjSHoqu+CgzjcxUUaPLMDXlG/LnkHxDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:17.712922Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.4269","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6bc00d4a7f54f05f0e907480527a56a80036b9e244966d21f20c7ce4da98fa03","sha256:b0efaa5c416cf5aef433749445d6b5d372c2a79e8bfbebb63003284198ce2acb"],"state_sha256":"104d6fe37bae448f0bc81fe12d1939cc76dc45c63cd6fd8e299279d89e186f46"}