{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:RRFNAQYBW7MR4L3UNQDMHAPA63","short_pith_number":"pith:RRFNAQYB","schema_version":"1.0","canonical_sha256":"8c4ad04301b7d91e2f746c06c381e0f6f1fa7d7405bdaf0db86834133196b4fc","source":{"kind":"arxiv","id":"1205.2321","version":2},"attestation_state":"computed","paper":{"title":"On the spectral density function of the Laplacian of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Herbert Koch, Wolfgang Lueck","submitted_at":"2012-05-10T17:34:19Z","abstract_excerpt":"Let X be a finite graph. Let E be the number of its edges and d be its degree. Denote by F_1(X) its first spectral density function which counts the number of eigenvalues less or equal to lambda^2 of the associated Laplace operator. We prove the estimate F_1(X)(lambda) - F_1(X)(0) le 2 cdot E cdot d cdot lambda for 0 le lambda < 1. We explain how this gives evidence for conjectures about approximating Fuglede-Kadison determinants and L^2-torsion."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1205.2321","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-10T17:34:19Z","cross_cats_sorted":[],"title_canon_sha256":"b7c0a7cdc3e81de42ef17628eb2402945cf9d77318221c291bdcefc88d815e4c","abstract_canon_sha256":"5229454c19a23b6478bd01158c3630fe4f17835e92c584f28709a4bf07b6107a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:39.603141Z","signature_b64":"ErQox5wgyXYBoUwjybvuCO+7guX6P8/mBbmneYRb9KUK/bIaae/rVe3cELXw1vTc998Yg/wbKnrTlzL6u46dAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c4ad04301b7d91e2f746c06c381e0f6f1fa7d7405bdaf0db86834133196b4fc","last_reissued_at":"2026-05-18T02:26:39.602736Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:39.602736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the spectral density function of the Laplacian of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Herbert Koch, Wolfgang Lueck","submitted_at":"2012-05-10T17:34:19Z","abstract_excerpt":"Let X be a finite graph. Let E be the number of its edges and d be its degree. Denote by F_1(X) its first spectral density function which counts the number of eigenvalues less or equal to lambda^2 of the associated Laplace operator. We prove the estimate F_1(X)(lambda) - F_1(X)(0) le 2 cdot E cdot d cdot lambda for 0 le lambda < 1. We explain how this gives evidence for conjectures about approximating Fuglede-Kadison determinants and L^2-torsion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.2321","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1205.2321","created_at":"2026-05-18T02:26:39.602793+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.2321v2","created_at":"2026-05-18T02:26:39.602793+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.2321","created_at":"2026-05-18T02:26:39.602793+00:00"},{"alias_kind":"pith_short_12","alias_value":"RRFNAQYBW7MR","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_16","alias_value":"RRFNAQYBW7MR4L3U","created_at":"2026-05-18T12:27:20.899486+00:00"},{"alias_kind":"pith_short_8","alias_value":"RRFNAQYB","created_at":"2026-05-18T12:27:20.899486+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63","json":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63.json","graph_json":"https://pith.science/api/pith-number/RRFNAQYBW7MR4L3UNQDMHAPA63/graph.json","events_json":"https://pith.science/api/pith-number/RRFNAQYBW7MR4L3UNQDMHAPA63/events.json","paper":"https://pith.science/paper/RRFNAQYB"},"agent_actions":{"view_html":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63","download_json":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63.json","view_paper":"https://pith.science/paper/RRFNAQYB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.2321&json=true","fetch_graph":"https://pith.science/api/pith-number/RRFNAQYBW7MR4L3UNQDMHAPA63/graph.json","fetch_events":"https://pith.science/api/pith-number/RRFNAQYBW7MR4L3UNQDMHAPA63/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63/action/storage_attestation","attest_author":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63/action/author_attestation","sign_citation":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63/action/citation_signature","submit_replication":"https://pith.science/pith/RRFNAQYBW7MR4L3UNQDMHAPA63/action/replication_record"}},"created_at":"2026-05-18T02:26:39.602793+00:00","updated_at":"2026-05-18T02:26:39.602793+00:00"}