{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:VI45J7ZOM4AXO3OKXQQ4KZMNDC","short_pith_number":"pith:VI45J7ZO","schema_version":"1.0","canonical_sha256":"aa39d4ff2e6701776dcabc21c5658d1888022f6767fdd1d0e0c5f0dbce98126a","source":{"kind":"arxiv","id":"1210.5736","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic enumeration of vertex-transitive graphs of fixed valency","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gabriel Verret, Pablo Spiga, Primoz Potocnik","submitted_at":"2012-10-21T15:07:35Z","abstract_excerpt":"Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \\emph{Cayley graph} $\\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\\log(CAY_d(n))\\in\\Theta (d(\\log n)^2)$, as $n\\to\\infty$. We also obtain some stronger results in the case $d=3$."},"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":"1210.5736","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-21T15:07:35Z","cross_cats_sorted":[],"title_canon_sha256":"c199a7b10996f9076b6daecbc0f3343a1d7f0b70b580978bfd4c44857dd98583","abstract_canon_sha256":"af7da35952f8fbc08689027399512c49b66a00d98d290637a2974000362f5f6d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:43.181158Z","signature_b64":"KXwkIDn6inMlOBqTOiQmL8dFcwD8tl5CoTjSXj74h/29PxUE4YmwNk52s4p/87aQgA9P/DnHDKmXCfBWmeyvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa39d4ff2e6701776dcabc21c5658d1888022f6767fdd1d0e0c5f0dbce98126a","last_reissued_at":"2026-05-18T03:42:43.180672Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:43.180672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic enumeration of vertex-transitive graphs of fixed valency","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gabriel Verret, Pablo Spiga, Primoz Potocnik","submitted_at":"2012-10-21T15:07:35Z","abstract_excerpt":"Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \\emph{Cayley graph} $\\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\\log(CAY_d(n))\\in\\Theta (d(\\log n)^2)$, as $n\\to\\infty$. We also obtain some stronger results in the case $d=3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5736","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":"1210.5736","created_at":"2026-05-18T03:42:43.180743+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.5736v1","created_at":"2026-05-18T03:42:43.180743+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5736","created_at":"2026-05-18T03:42:43.180743+00:00"},{"alias_kind":"pith_short_12","alias_value":"VI45J7ZOM4AX","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_16","alias_value":"VI45J7ZOM4AXO3OK","created_at":"2026-05-18T12:27:25.539911+00:00"},{"alias_kind":"pith_short_8","alias_value":"VI45J7ZO","created_at":"2026-05-18T12:27:25.539911+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/VI45J7ZOM4AXO3OKXQQ4KZMNDC","json":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC.json","graph_json":"https://pith.science/api/pith-number/VI45J7ZOM4AXO3OKXQQ4KZMNDC/graph.json","events_json":"https://pith.science/api/pith-number/VI45J7ZOM4AXO3OKXQQ4KZMNDC/events.json","paper":"https://pith.science/paper/VI45J7ZO"},"agent_actions":{"view_html":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC","download_json":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC.json","view_paper":"https://pith.science/paper/VI45J7ZO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.5736&json=true","fetch_graph":"https://pith.science/api/pith-number/VI45J7ZOM4AXO3OKXQQ4KZMNDC/graph.json","fetch_events":"https://pith.science/api/pith-number/VI45J7ZOM4AXO3OKXQQ4KZMNDC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC/action/storage_attestation","attest_author":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC/action/author_attestation","sign_citation":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC/action/citation_signature","submit_replication":"https://pith.science/pith/VI45J7ZOM4AXO3OKXQQ4KZMNDC/action/replication_record"}},"created_at":"2026-05-18T03:42:43.180743+00:00","updated_at":"2026-05-18T03:42:43.180743+00:00"}