{"paper":{"title":"Squared distance matrix of a weighted tree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ravindra B. Bapat","submitted_at":"2018-10-15T05:08:58Z","abstract_excerpt":"Let $T$ be a tree with vertex set $\\{1, \\ldots, n\\}$ such that each edge is assigned a nonzero weight. The squared distance matrix of $T,$ denoted by $\\Delta,$ is the $n \\times n$ matrix with $(i,j)$-element $d(i,j)^2,$ where $d(i,j)$ is the sum of the weights of the edges on the $(ij)$-path. We obtain a formula for the determinant of $\\Delta.$ A formula for $\\Delta^{-1}$ is also obtained, under certain conditions. The results generalize known formulas for the unweighted case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06182","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}