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Suppose that $m,n,m',n'\\geq2$ are integers and $\\mathbb{D}'$ is a division ring. Using the weighted semi-affine map and algebraic method, we characterize graph homomorphisms from ${\\mathbb{D}}^{m\\times n}$ to ${\\mathbb{D}'}^{m'\\times n'}$ (where $|\\mathbb{D}|\\geq 4$) under some weaker conditions."},"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":"1701.07324","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-25T14:15:42Z","cross_cats_sorted":[],"title_canon_sha256":"5a0a3158c042a07561204e19079dc1f6d631a0354c8a94d93d05c5a2dafdcf78","abstract_canon_sha256":"2a47a34d9074dc06c738c42338103dceea6a7f04206ae0c3130d918cc0f1aa14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:13.433924Z","signature_b64":"1joY87w7mt5CEuL9T+ZWCrpsTaEblzqAfjw/5cE0ujMYLzg5vmpx76JjJqfm91m8OXIs1AkRLGgXLVZt/y+uAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"975ece718fb6b3d265973c91f8ea2c5deef4657dc23055aca08e216f556a80c7","last_reissued_at":"2026-05-18T00:44:13.433372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:13.433372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Graph homomorphisms on rectangular matrices over division rings I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kang Zhao, Li-Ping Huang","submitted_at":"2017-01-25T14:15:42Z","abstract_excerpt":"Let $\\mathbb{D}$ be a division ring, and let ${\\mathbb{D}}^{m\\times n}$ be the set of $m\\times n$ matrices over $\\mathbb{D}$. 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