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The vertex set $V(\\Gamma(G))$ of $\\Gamma(G)$ is $G\\setminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if and only if $xy\\neq yx$. For non--abelian finite groups $G$ and $H$ it is conjectured that if $\\Gamma(G) \\cong \\Gamma(H)$, then $|G|=|H|$. We prove the conjecture."},"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":"1706.07142","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-06-22T00:12:51Z","cross_cats_sorted":[],"title_canon_sha256":"58fe71ce50803e194caf02108987aa654bf038101d19c9a40970bd69de4d5854","abstract_canon_sha256":"7a57b6cc39dd967e678fecd4ff6b5418e485970c1b4ad72f66069cf0db589fe5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:41.478928Z","signature_b64":"S5PFg7566kpCLBjYWwz+fzG5ppsZNL16Vhtg46MUAe4t923c1cxzszETAcNartHe3W+djfwbUCisqTLi27tcCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a0dbca78e6048120188c20c6b0240bd59044743e93bb085f78a3b5e769cd0bf","last_reissued_at":"2026-05-18T00:34:41.478307Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:41.478307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a conjecture of Abdollahi-Akbari-Maimani concerning the non-commutative graph of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Armando S\\'anchez-Nungaray, Daniel G. Mendoza, Luis A. Dupont","submitted_at":"2017-06-22T00:12:51Z","abstract_excerpt":"The non--commuting graph $\\Gamma(G)$ of a non--abelian group $G$ is defined as follows. The vertex set $V(\\Gamma(G))$ of $\\Gamma(G)$ is $G\\setminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if and only if $xy\\neq yx$. For non--abelian finite groups $G$ and $H$ it is conjectured that if $\\Gamma(G) \\cong \\Gamma(H)$, then $|G|=|H|$. 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