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We obtain that $$\\det[x+d(v_{j+1},v_k)]_{1\\le j,k\\le n}=2^{n-2}\\prod_{e\\in E(T)}w(e),$$ where $d(v_{j+1},v_k)$ is the weighted distance between $v_{j+1}$ and $v_k$ in the tree $T$. This is similar to the celebrated Graham-Pollak theorem on determinants of distance matrices for trees. Actually, a more general result is deduced in this paper."},"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":"2303.12629","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-03-22T15:10:26Z","cross_cats_sorted":[],"title_canon_sha256":"f801e2ee10f82156178979d335b614716d7aacb13928ba8ea87c58969fc4d6e2","abstract_canon_sha256":"13c033caefb98ce9ff40df405fea36b51913c53f7585a8c0d1f9e2a199086990"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:58:20.223209Z","signature_b64":"HFML48LmtFkgupsYQC21NpyPSZSTOP8c9d/a8l2LhuxyO3GQL3+zXu8aLXN+ZIuaAmVree6WfwZ7JDO7acOTBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7eaf820fd983982eb6370d1fc548c82fa3e094b6e51ef8cfc55275a4759d7860","last_reissued_at":"2026-07-05T05:58:20.222809Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:58:20.222809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new result similar to the Graham-Pollak theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhi-Wei Sun","submitted_at":"2023-03-22T15:10:26Z","abstract_excerpt":"Let $n>1$ be an integer, and let $T$ be a tree with $n+1$ vertices $v_1,\\ldots,v_{n+1}$, where $v_1$ and $v_{n+1}$ are two leaves of $T$. For each edge $e$ of $T$, assign a complex number $w(e)$ as its weight. We obtain that $$\\det[x+d(v_{j+1},v_k)]_{1\\le j,k\\le n}=2^{n-2}\\prod_{e\\in E(T)}w(e),$$ where $d(v_{j+1},v_k)$ is the weighted distance between $v_{j+1}$ and $v_k$ in the tree $T$. This is similar to the celebrated Graham-Pollak theorem on determinants of distance matrices for trees. 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