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Ajesh Babu","submitted_at":"2010-11-04T03:48:37Z","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)^{"},"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":"1011.1058","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2010-11-04T03:48:37Z","cross_cats_sorted":[],"title_canon_sha256":"b6b805d058ecc31347e875166d5541ccb97f5dbc8113642ffbd2e77c4f9df672","abstract_canon_sha256":"b1957d2cbaf36556df1a55525a04b42d6d3b27634a18427c4a1381c01967e3cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:37:09.707610Z","signature_b64":"3qikYPwKp1KG3x1PD68oMW59Gm47pv46og+8ECNZZmpoQoOcrlsV+QSVmRt73u2lhIK4cRbWzP1MF6+7I9MPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eef7518e3f900760d4041824c29416970fe4c387a97a9695beff7de984af5765","last_reissued_at":"2026-05-18T04:37:09.707143Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:37:09.707143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An entropy based proof of the Moore bound for irregular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Jaikumar Radhakrishnan, S. Ajesh Babu","submitted_at":"2010-11-04T03:48:37Z","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)^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1058","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":"1011.1058","created_at":"2026-05-18T04:37:09.707211+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.1058v1","created_at":"2026-05-18T04:37:09.707211+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1058","created_at":"2026-05-18T04:37:09.707211+00:00"},{"alias_kind":"pith_short_12","alias_value":"533VDDR7SADW","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"533VDDR7SADWBVAE","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"533VDDR7","created_at":"2026-05-18T12:26:04.259169+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/533VDDR7SADWBVAEDASMFFAWS4","json":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4.json","graph_json":"https://pith.science/api/pith-number/533VDDR7SADWBVAEDASMFFAWS4/graph.json","events_json":"https://pith.science/api/pith-number/533VDDR7SADWBVAEDASMFFAWS4/events.json","paper":"https://pith.science/paper/533VDDR7"},"agent_actions":{"view_html":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4","download_json":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4.json","view_paper":"https://pith.science/paper/533VDDR7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.1058&json=true","fetch_graph":"https://pith.science/api/pith-number/533VDDR7SADWBVAEDASMFFAWS4/graph.json","fetch_events":"https://pith.science/api/pith-number/533VDDR7SADWBVAEDASMFFAWS4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4/action/storage_attestation","attest_author":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4/action/author_attestation","sign_citation":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4/action/citation_signature","submit_replication":"https://pith.science/pith/533VDDR7SADWBVAEDASMFFAWS4/action/replication_record"}},"created_at":"2026-05-18T04:37:09.707211+00:00","updated_at":"2026-05-18T04:37:09.707211+00:00"}