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In this note, we show that the {\\it oriented} transition matrices over the field $\\mathcal R$ of all real numbers (over the finite field $\\mathcal Z_2$ of two elements respectively) of all continuous {\\it vertex maps} on {\\it all} oriented trees with $n+1$ vertices are similar to one another over $\\mathcal R$ (over $\\mathcal Z_2$ respectively) and have characteristic polynomial $\\sum_{k=0}^n x^k$. Consequently, the {\\it unoriented} transition matrices over the field $Z_2$ of all continuous {\\it vertex maps} on {\\it all} oriented trees with $n+1$ vertices are simila"},"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":"1503.04568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-03-16T08:29:34Z","cross_cats_sorted":[],"title_canon_sha256":"48559b04a3ae6bc914e608440949e09ed213f3c3c7c5344e8d12982e46904d1a","abstract_canon_sha256":"3217ae4773c139dc70e7540a6fb7e23bb9c4a1758f5e8a1afc3f1f424f006bd4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:23:24.662404Z","signature_b64":"oZc+/glnuLEUQXJoPDtCtoctWtnSpL5OSHH4i4oUgFquoMUuVrM991iqDOE0v4/pL9a2Te0OnLNH6/1tkZZJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"456a092167d69827d5a75dacaa438750cde30c638ee5501c94d30e01c464cd2f","last_reissued_at":"2026-05-18T02:23:24.661769Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:23:24.661769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Class of Similar Square {-1,0,1}-Matrices Arising from Vertex maps on Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bau-Sen Du","submitted_at":"2015-03-16T08:29:34Z","abstract_excerpt":"Let $n \\ge 2$ be an integer. 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