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In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\\it\\Gamma$, either the full automorphism group ${\\sf Aut}\\it\\Gamma$ is isomorphic to ${\\sf PSL}(2,p)$, ${\\sf PGL}(2,p)$, ${\\sf PSL}(2,p){\\"},"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":"1702.05750","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-19T13:27:51Z","cross_cats_sorted":[],"title_canon_sha256":"fff36f32b1e7ecc144cbec6080711c56339b51cd6a1dde3543a6e8f0f4218070","abstract_canon_sha256":"e21f3f91634bea2ff6502c97ef5c2aed55b0c33c8586590851e2ad4aa22084e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:25.419535Z","signature_b64":"P70nm/XFcM93VDNIz7O2E8yec54MR84ZVW7x+6WEV8UHAB+nDl5S3Qg11i0IGwo7/c560Wc4jyv/uvIQ29V0AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b815401b2d1b5271b08ccf1e03e82869d8c86a7110e28f7b028e8647e9ef15e4","last_reissued_at":"2026-05-18T00:50:25.418853Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:25.418853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pentavalent symmetric graphs of order four times an odd square-free integer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Gong Lou, Bo Ling, Ci Xuan Wu","submitted_at":"2017-02-19T13:27:51Z","abstract_excerpt":"A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \\#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \\#P36, 2013) determined all pentavalent symmetric graphs of order $4pq$. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\\it\\Gamma$, either the full automorphism group ${\\sf Aut}\\it\\Gamma$ is isomorphic to ${\\sf PSL}(2,p)$, ${\\sf PGL}(2,p)$, ${\\sf PSL}(2,p){\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.05750","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":"1702.05750","created_at":"2026-05-18T00:50:25.418945+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.05750v1","created_at":"2026-05-18T00:50:25.418945+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.05750","created_at":"2026-05-18T00:50:25.418945+00:00"},{"alias_kind":"pith_short_12","alias_value":"XAKUAGZNDNJH","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"XAKUAGZNDNJHDMEM","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"XAKUAGZN","created_at":"2026-05-18T12:31:53.515858+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/XAKUAGZNDNJHDMEMZ4PAH2BINH","json":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH.json","graph_json":"https://pith.science/api/pith-number/XAKUAGZNDNJHDMEMZ4PAH2BINH/graph.json","events_json":"https://pith.science/api/pith-number/XAKUAGZNDNJHDMEMZ4PAH2BINH/events.json","paper":"https://pith.science/paper/XAKUAGZN"},"agent_actions":{"view_html":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH","download_json":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH.json","view_paper":"https://pith.science/paper/XAKUAGZN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.05750&json=true","fetch_graph":"https://pith.science/api/pith-number/XAKUAGZNDNJHDMEMZ4PAH2BINH/graph.json","fetch_events":"https://pith.science/api/pith-number/XAKUAGZNDNJHDMEMZ4PAH2BINH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH/action/storage_attestation","attest_author":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH/action/author_attestation","sign_citation":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH/action/citation_signature","submit_replication":"https://pith.science/pith/XAKUAGZNDNJHDMEMZ4PAH2BINH/action/replication_record"}},"created_at":"2026-05-18T00:50:25.418945+00:00","updated_at":"2026-05-18T00:50:25.418945+00:00"}