{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:533VDDR7SADWBVAEDASMFFAWS4","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":"b1957d2cbaf36556df1a55525a04b42d6d3b27634a18427c4a1381c01967e3cf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-11-04T03:48:37Z","title_canon_sha256":"b6b805d058ecc31347e875166d5541ccb97f5dbc8113642ffbd2e77c4f9df672"},"schema_version":"1.0","source":{"id":"1011.1058","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1058","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1058v1","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1058","created_at":"2026-05-18T04:37:09Z"},{"alias_kind":"pith_short_12","alias_value":"533VDDR7SADW","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"533VDDR7SADWBVAE","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"533VDDR7","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:6f5e32766c118e7db07df22488ef0a504d175c3097f5e14e32d5ef43a98d1dd5","target":"graph","created_at":"2026-05-18T04:37:09Z","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":"We provide proofs of the following theorems by considering the entropy of random walks: Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple graph with n vertices, girth g, minimum degree at least 2 and average degree d: Odd girth: If g=2r+1,then n \\geq 1 + d*(\\Sum_{i=0}^{r-1}(d-1)^i) Even girth: If g=2r,then n \\geq 2*(\\Sum_{i=0}^{r-1} (d-1)^i) Theorem 2.(Hoory) Let G = (V_L,V_R,E) be a bipartite graph of girth g = 2r, with n_L = |V_L| and n_R = |V_R|, minimum degree at least 2 and the left and right average degrees d_L and d_R. Then, n_L \\geq \\Sum_{i=0}^{r-1}(d_R-1)^{i/2}(d_L-1)^{","authors_text":"Jaikumar Radhakrishnan, S. Ajesh Babu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-11-04T03:48:37Z","title":"An entropy based proof of the Moore bound for irregular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1058","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":"sha256:842a35c5e0a549c2fd1ce6cdbda3d13cc2323702e5b836c9ba4971b396fd2b51","target":"record","created_at":"2026-05-18T04:37:09Z","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":"b1957d2cbaf36556df1a55525a04b42d6d3b27634a18427c4a1381c01967e3cf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-11-04T03:48:37Z","title_canon_sha256":"b6b805d058ecc31347e875166d5541ccb97f5dbc8113642ffbd2e77c4f9df672"},"schema_version":"1.0","source":{"id":"1011.1058","kind":"arxiv","version":1}},"canonical_sha256":"eef7518e3f900760d4041824c29416970fe4c387a97a9695beff7de984af5765","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eef7518e3f900760d4041824c29416970fe4c387a97a9695beff7de984af5765","first_computed_at":"2026-05-18T04:37:09.707143Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:37:09.707143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3qikYPwKp1KG3x1PD68oMW59Gm47pv46og+8ECNZZmpoQoOcrlsV+QSVmRt73u2lhIK4cRbWzP1MF6+7I9MPCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:37:09.707610Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.1058","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:842a35c5e0a549c2fd1ce6cdbda3d13cc2323702e5b836c9ba4971b396fd2b51","sha256:6f5e32766c118e7db07df22488ef0a504d175c3097f5e14e32d5ef43a98d1dd5"],"state_sha256":"c48216ebfb5f60b5b3008a3bb351135ebf9011181b3dfce30cfd378c8da0c63f"}