{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:AQ7FPHKWMXPW7KZ3B2SK2HZUJQ","short_pith_number":"pith:AQ7FPHKW","schema_version":"1.0","canonical_sha256":"043e579d5665df6fab3b0ea4ad1f344c16a4c7a8112807e695ed2feb4ef57199","source":{"kind":"arxiv","id":"1808.06101","version":1},"attestation_state":"computed","paper":{"title":"Spanning tree packing, edge-connectivity and eigenvalues of graphs with given girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong-Jian Lai, Ruifang Liu, Yingzhi Tian","submitted_at":"2018-08-18T16:37:49Z","abstract_excerpt":"Let $\\tau(G)$ and $\\kappa'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $\\delta \\ge 2k \\ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $\\lambda_2(G) < \\delta - \\frac{2k-1}{\\delta+1}$, then $\\tau(G) \\ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(\\delta, k, g)$ such that for every gra"},"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":"1808.06101","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-18T16:37:49Z","cross_cats_sorted":[],"title_canon_sha256":"56090ed64e6c67608de657f55fafb7757b5011b9a810179dd1f9b8ea4ebdc4ee","abstract_canon_sha256":"0c8c21a9ba3814cfcda908388dca851a93bbb330a5794e807403901a5db69534"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:46.104636Z","signature_b64":"9vJti2qZo5rReAs+ywlLiS2dkvd1SNiTAtvE6Fu7YaaIkKL8RbdGc6oP36RKuqCJSRAZpeJDGs6LD57GaJjzAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"043e579d5665df6fab3b0ea4ad1f344c16a4c7a8112807e695ed2feb4ef57199","last_reissued_at":"2026-05-18T00:07:46.104009Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:46.104009Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spanning tree packing, edge-connectivity and eigenvalues of graphs with given girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong-Jian Lai, Ruifang Liu, Yingzhi Tian","submitted_at":"2018-08-18T16:37:49Z","abstract_excerpt":"Let $\\tau(G)$ and $\\kappa'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $\\delta \\ge 2k \\ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $\\lambda_2(G) < \\delta - \\frac{2k-1}{\\delta+1}$, then $\\tau(G) \\ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(\\delta, k, g)$ such that for every gra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06101","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":"1808.06101","created_at":"2026-05-18T00:07:46.104116+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.06101v1","created_at":"2026-05-18T00:07:46.104116+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.06101","created_at":"2026-05-18T00:07:46.104116+00:00"},{"alias_kind":"pith_short_12","alias_value":"AQ7FPHKWMXPW","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"AQ7FPHKWMXPW7KZ3","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"AQ7FPHKW","created_at":"2026-05-18T12:32:13.499390+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/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ","json":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ.json","graph_json":"https://pith.science/api/pith-number/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/graph.json","events_json":"https://pith.science/api/pith-number/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/events.json","paper":"https://pith.science/paper/AQ7FPHKW"},"agent_actions":{"view_html":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ","download_json":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ.json","view_paper":"https://pith.science/paper/AQ7FPHKW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.06101&json=true","fetch_graph":"https://pith.science/api/pith-number/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/graph.json","fetch_events":"https://pith.science/api/pith-number/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/action/storage_attestation","attest_author":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/action/author_attestation","sign_citation":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/action/citation_signature","submit_replication":"https://pith.science/pith/AQ7FPHKWMXPW7KZ3B2SK2HZUJQ/action/replication_record"}},"created_at":"2026-05-18T00:07:46.104116+00:00","updated_at":"2026-05-18T00:07:46.104116+00:00"}