{"paper":{"title":"The number of trees in a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dhruv Mubayi, Jacques Verstraete","submitted_at":"2015-11-23T15:31:31Z","abstract_excerpt":"Let $T$ be a tree with $t$ edges. We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \\[2|E| \\prod_{v \\in V} (d(v) - t + 1)^{\\frac{(t-1)d(v)}{2|E|}}.\\] Consequently, any $n$-vertex graph of average degree $d$ and minimum degree at least $t$ contains at least\n  $$nd(d-t+1)^{t-1}$$ isomorphic (labeled) copies of $T$.\n  This answers a question of Dellamonica et. al. (where the above statement was proved when $T$ is the path with three edges) while extending an old result of Erd\\H os and Simonovits."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07274","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"}