{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:SLDLN52G54HEYBCKSOBI3EUFYX","short_pith_number":"pith:SLDLN52G","schema_version":"1.0","canonical_sha256":"92c6b6f746ef0e4c044a93828d9285c5ff60e20313a78461b62eac0dca88fd22","source":{"kind":"arxiv","id":"1708.07979","version":1},"attestation_state":"computed","paper":{"title":"Graphs with at most three distance eigenvalues different from $-1$ and $-2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lu Lu, Qiongxiang Huang, Xueyi Huang","submitted_at":"2017-08-26T14:42:14Z","abstract_excerpt":"Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distance matrix of $G$. Let $\\partial_1(G)\\ge\\partial_2(G)\\ge\\cdots\\ge\\partial_n(G)$ denote the eigenvalues of $D(G)$. In this paper, we characterize all connected graphs with $\\partial_{3}(G)\\leq -1$ and $\\partial_{n-1}(G)\\geq -2$. By the way, we determine all connected graphs with at most three distance eigenvalues different from $-1$ and $-2$."},"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":"1708.07979","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-26T14:42:14Z","cross_cats_sorted":[],"title_canon_sha256":"80ce920e7dfa6f1e09de0adc1066b4bf0a905e2761ac8b8c07b855ce4a75f138","abstract_canon_sha256":"d69a6d2cabc44a107d561111f6208a19890975c7bed1e6ee94d8fcf9cb8d4403"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:37.413972Z","signature_b64":"SpILLdbuQ0TQGqxSmmpTzHRkm9xMj7Mp62z1GlRsUOzY62GbSLPr8kUfLedNefbWKYI0hYndxz32QDe78D9HCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92c6b6f746ef0e4c044a93828d9285c5ff60e20313a78461b62eac0dca88fd22","last_reissued_at":"2026-05-18T00:36:37.413138Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:37.413138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Graphs with at most three distance eigenvalues different from $-1$ and $-2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lu Lu, Qiongxiang Huang, Xueyi Huang","submitted_at":"2017-08-26T14:42:14Z","abstract_excerpt":"Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distance matrix of $G$. Let $\\partial_1(G)\\ge\\partial_2(G)\\ge\\cdots\\ge\\partial_n(G)$ denote the eigenvalues of $D(G)$. In this paper, we characterize all connected graphs with $\\partial_{3}(G)\\leq -1$ and $\\partial_{n-1}(G)\\geq -2$. By the way, we determine all connected graphs with at most three distance eigenvalues different from $-1$ and $-2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.07979","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1708.07979","created_at":"2026-05-18T00:36:37.413276+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.07979v1","created_at":"2026-05-18T00:36:37.413276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.07979","created_at":"2026-05-18T00:36:37.413276+00:00"},{"alias_kind":"pith_short_12","alias_value":"SLDLN52G54HE","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_16","alias_value":"SLDLN52G54HEYBCK","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_8","alias_value":"SLDLN52G","created_at":"2026-05-18T12:31:43.269735+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/SLDLN52G54HEYBCKSOBI3EUFYX","json":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX.json","graph_json":"https://pith.science/api/pith-number/SLDLN52G54HEYBCKSOBI3EUFYX/graph.json","events_json":"https://pith.science/api/pith-number/SLDLN52G54HEYBCKSOBI3EUFYX/events.json","paper":"https://pith.science/paper/SLDLN52G"},"agent_actions":{"view_html":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX","download_json":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX.json","view_paper":"https://pith.science/paper/SLDLN52G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.07979&json=true","fetch_graph":"https://pith.science/api/pith-number/SLDLN52G54HEYBCKSOBI3EUFYX/graph.json","fetch_events":"https://pith.science/api/pith-number/SLDLN52G54HEYBCKSOBI3EUFYX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX/action/storage_attestation","attest_author":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX/action/author_attestation","sign_citation":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX/action/citation_signature","submit_replication":"https://pith.science/pith/SLDLN52G54HEYBCKSOBI3EUFYX/action/replication_record"}},"created_at":"2026-05-18T00:36:37.413276+00:00","updated_at":"2026-05-18T00:36:37.413276+00:00"}