{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3CIORN5VZJFK44AAZMHUV3RKDF","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":"9e5039f7e036e72369f23984c2e3519c9bee61e244ae940c3e7254eef21c21d9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-10T17:32:53Z","title_canon_sha256":"83ca1c35a8b096c7307c246c1b5207d91a32ef61cf5f4fa238d991500c5395dd"},"schema_version":"1.0","source":{"id":"2608.09879","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.09879","created_at":"2026-08-11T02:25:19Z"},{"alias_kind":"arxiv_version","alias_value":"2608.09879v1","created_at":"2026-08-11T02:25:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.09879","created_at":"2026-08-11T02:25:19Z"},{"alias_kind":"pith_short_12","alias_value":"3CIORN5VZJFK","created_at":"2026-08-11T02:25:19Z"},{"alias_kind":"pith_short_16","alias_value":"3CIORN5VZJFK44AA","created_at":"2026-08-11T02:25:19Z"},{"alias_kind":"pith_short_8","alias_value":"3CIORN5V","created_at":"2026-08-11T02:25:19Z"}],"graph_snapshots":[{"event_id":"sha256:4b039b93b4ec96a4ee53436c2a42add95cd2de5a16aef66b98855cef584fae70","target":"graph","created_at":"2026-08-11T02:25:19Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.09879/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The algebraic connectivity of a graph $G$ is a well studied graph invariant that is related to other properties of the graph such as connectivity and expansion. Given $n$ and $m$, $\\alpha(n,m)$ is the maximum algebraic connectivity of a graph with $n$ vertices with $m$ edges. In 2015, Kolokolnikov conjectured that $\\alpha(n,2n-4)=2$ for $n\\geq 3$, and verified this claim computationally for $n \\le 12$. In this paper, we prove Kolokolnikov's conjecture.","authors_text":"Abhay Jayarajan, M. Rajesh Kannan, Rahul Roy, Sebastian M. Cioab\\u{a}","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-10T17:32:53Z","title":"Maximizing the algebraic connectivity of graphs of given order and size: a proof of a conjecture of Kolokolnikov"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09879","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":"sha256:4b71ddc8f7f1b2dca80a38747a704e8961966aa80417c4232625c92ca8eecec3","target":"record","created_at":"2026-08-11T02:25:19Z","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":"9e5039f7e036e72369f23984c2e3519c9bee61e244ae940c3e7254eef21c21d9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-10T17:32:53Z","title_canon_sha256":"83ca1c35a8b096c7307c246c1b5207d91a32ef61cf5f4fa238d991500c5395dd"},"schema_version":"1.0","source":{"id":"2608.09879","kind":"arxiv","version":1}},"canonical_sha256":"d890e8b7b5ca4aae7000cb0f4aee2a1941dff233d786db32792e11983b7d42b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d890e8b7b5ca4aae7000cb0f4aee2a1941dff233d786db32792e11983b7d42b2","first_computed_at":"2026-08-11T02:25:19.432041Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T02:25:19.432041Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9Y83y43udLULi6KKzZG2XJJq1vEP1oJlC1okKQCe4GFpRbpUMU9wfU+j3zLaX2HOKBzxYihGR8vQFaNbkR6RDg==","signature_status":"signed_v1","signed_at":"2026-08-11T02:25:19.433617Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.09879","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b71ddc8f7f1b2dca80a38747a704e8961966aa80417c4232625c92ca8eecec3","sha256:4b039b93b4ec96a4ee53436c2a42add95cd2de5a16aef66b98855cef584fae70"],"state_sha256":"314b05b12d3cba2764077eb128a0a029a26b13e6388d81e80a80a9a68311f655"}