{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AQ7FPHKWMXPW7KZ3B2SK2HZUJQ","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":"0c8c21a9ba3814cfcda908388dca851a93bbb330a5794e807403901a5db69534","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-18T16:37:49Z","title_canon_sha256":"56090ed64e6c67608de657f55fafb7757b5011b9a810179dd1f9b8ea4ebdc4ee"},"schema_version":"1.0","source":{"id":"1808.06101","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.06101","created_at":"2026-05-18T00:07:46Z"},{"alias_kind":"arxiv_version","alias_value":"1808.06101v1","created_at":"2026-05-18T00:07:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.06101","created_at":"2026-05-18T00:07:46Z"},{"alias_kind":"pith_short_12","alias_value":"AQ7FPHKWMXPW","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AQ7FPHKWMXPW7KZ3","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AQ7FPHKW","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:17c4b24e0d490add0bb265b64c70ec3d25b4131795194f1b83a0e1937adfcb07","target":"graph","created_at":"2026-05-18T00:07:46Z","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":"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","authors_text":"Hong-Jian Lai, Ruifang Liu, Yingzhi Tian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-18T16:37:49Z","title":"Spanning tree packing, edge-connectivity and eigenvalues of graphs with given girth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06101","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":"sha256:15e1ab0cc500898bc30d52b0e6bd5fdc026f5ab0bb64a0736d33010dd97c0a4f","target":"record","created_at":"2026-05-18T00:07:46Z","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":"0c8c21a9ba3814cfcda908388dca851a93bbb330a5794e807403901a5db69534","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-18T16:37:49Z","title_canon_sha256":"56090ed64e6c67608de657f55fafb7757b5011b9a810179dd1f9b8ea4ebdc4ee"},"schema_version":"1.0","source":{"id":"1808.06101","kind":"arxiv","version":1}},"canonical_sha256":"043e579d5665df6fab3b0ea4ad1f344c16a4c7a8112807e695ed2feb4ef57199","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"043e579d5665df6fab3b0ea4ad1f344c16a4c7a8112807e695ed2feb4ef57199","first_computed_at":"2026-05-18T00:07:46.104009Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:46.104009Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9vJti2qZo5rReAs+ywlLiS2dkvd1SNiTAtvE6Fu7YaaIkKL8RbdGc6oP36RKuqCJSRAZpeJDGs6LD57GaJjzAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:46.104636Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.06101","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15e1ab0cc500898bc30d52b0e6bd5fdc026f5ab0bb64a0736d33010dd97c0a4f","sha256:17c4b24e0d490add0bb265b64c70ec3d25b4131795194f1b83a0e1937adfcb07"],"state_sha256":"5c8ee2dad65ca59695c91615c027ddfa84127b0a97e6e1f79d885c9cca60066b"}