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The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that lie on adjacent edges of G and the central graph C(G) of G is obtained from S(G) after joining all non-adjacent vertices of G.\n  We show that if the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups) and apply this result to obtain new upper bounds of the distinguishing number and the distinguishing index of C(G) and M(G) and provide examples showing that these bounds c","authors_text":"Alexa Gopaulsingh, Amitayu Banerjee, Zal\\'an Moln\\'ar","cross_cats":["math.GR"],"headline":"If a graph G has at least four vertices, then Aut(G) is isomorphic to Aut(C(G)) and Aut(M(G)).","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-22T07:39:10Z","title":"Automorphism groups and Distinguishing Colorings of Central and Middle Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.16301","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-19T03:50:21.418463Z","id":"9d45486b-06b0-4481-9304-ef39905a88ca","model_set":{"reader":"grok-4.3"},"one_line_summary":"For connected graphs G with |V(G)| >= 4, Aut(G) ≅ Aut(C(G)) ≅ Aut(M(G)) as abstract groups, which yields new upper bounds on the distinguishing number and index of the central and middle graphs.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"If a graph G has at least four vertices, then Aut(G) is isomorphic to Aut(C(G)) and Aut(M(G)).","strongest_claim":"If the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups).","weakest_assumption":"The constructions of C(G) and M(G) from the subdivision graph S(G) preserve the full automorphism group of G when |V(G)| >= 4; this is the load-bearing step that allows the isomorphism to be claimed and the distinguishing bounds to follow."}},"verdict_id":"9d45486b-06b0-4481-9304-ef39905a88ca"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:65b7a98810886fad0e72b71a6f75bb7b34fa17f9196c3096ad433c0ceaccbb75","target":"record","created_at":"2026-06-19T16:09:50Z","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":"36f2f34241c1c5dfca22ef9d9e1a3da3d0806ae8f0daa0daea0d18afc9164bfe","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-22T07:39:10Z","title_canon_sha256":"18503ca3409ccc39404238e958c3a7223a99712a9970408280e7ef88b9159252"},"schema_version":"1.0","source":{"id":"2507.16301","kind":"arxiv","version":3}},"canonical_sha256":"f0b832fe4c0b37a7bcebb41ba49f7fbfccff0d8db2dcbf4ec6d75c2082b71b6f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f0b832fe4c0b37a7bcebb41ba49f7fbfccff0d8db2dcbf4ec6d75c2082b71b6f","first_computed_at":"2026-06-19T16:09:50.351496Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:09:50.351496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"847AF6HtJscZAj++aLr4mxVE25svQ2EP42GKLbKXV3uMJ6eqRfMd6PW08DN4dy4gR8OzhGru1qXZvf2H5UxfCA==","signature_status":"signed_v1","signed_at":"2026-06-19T16:09:50.351926Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.16301","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65b7a98810886fad0e72b71a6f75bb7b34fa17f9196c3096ad433c0ceaccbb75","sha256:46721e14368cf90f119ddf537d142599c9f38d5df397fbb15351ed626e6cdb85"],"state_sha256":"7f83ded749bca3792503152207e354005d9606477c2c1482f0ebfe9bd96e0cab"}