{"paper":{"title":"A formula for the number of the spanning trees of line graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Helin Gong, Xian'an Jin","submitted_at":"2015-07-23T05:50:15Z","abstract_excerpt":"Let $G=(V,E)$ be a loopless graph and $\\mathcal{T}(G)$ be the set of all spanning trees of $G$. Let $L(G)$ be the line graph of the graph $G$ and $t(L(G))$ be the number of spanning trees of $L(G)$. Then, by using techniques from electrical networks, we obtain the following formula: $$ t(L(G)) = \\frac{1}{\\prod_{v\\in V}d^2(v)}\\sum_{T\\subseteq \\mathcal{T}(G)}\\big[\\prod_{e = xy\\in T}d(x)d(y)\\big]\\big[\\prod_{e = uv\\in E\\backslash T}[d(u)+d(v)]\\big]. $$ As a result, we provide a very simple and different proof of the formula on the number of spanning trees of some irregular line graphs, and give a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06389","kind":"arxiv","version":2},"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"}