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We show that there are infinitely many geodesic transitive graphs with this property for each of these values of $s$, and that these graphs can have arbitrarily large diameter if and only if $1\\leq s\\leq 3$. Moreover, for a prime $p$ we prove that there exists a graph of valency $p$ that is 2-geodesic trans"},"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":"1110.2235","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-11T00:05:54Z","cross_cats_sorted":[],"title_canon_sha256":"d2d968554af98cb0d5fb11bd877eaa4a4fa49e3f386e5650b086be21bac716e2","abstract_canon_sha256":"e6fd655eec171f49f8c8df2e8137749cc974237bd3108678af48732eeced351e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:17.288575Z","signature_b64":"Y0Ygi/6m8sZppEJAw1wj2Sf60VTCce+XcNVXMiTWXJ4wdTAXJaK15uza+ZEENx7bweJaw9tRVWIoe89inAICCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9259230d453707fcff330d6208364df0ae9b805449a697e962347ce2afbaeed","last_reissued_at":"2026-05-18T04:11:17.288032Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:17.288032Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On distance, geodesic and arc transitivity of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alice Devillers, Cai Heng Li, Cheryl E. 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