{"paper":{"title":"Sharp upper and lower bounds on the number of spanning trees in Cartesian product of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jernej Azarija","submitted_at":"2012-10-23T19:47:15Z","abstract_excerpt":"Let $G_1$ and $G_2$ be simple graphs and let $n_1 = |V(G_1)|$, $m_1 = |E(G_1)|$, $n_2 = |V(G_2)|$ and $m_2 = |E(G_2)|.$ In this paper we derive sharp upper and lower bounds for the number of spanning trees $\\tau$ in the Cartesian product $G_1 \\square G_2$ of $G_1$ and $G_2$. We show that: $$ \\tau(G_1 \\square G_2) \\geq \\frac{2^{(n_1-1)(n_2-1)}}{n_1n_2} (\\tau(G_1) n_1)^{\\frac{n_2+1}{2}} (\\tau(G_2)n_2)^{\\frac{n_1+1}{2}}$$ and $$\\tau(G_1 \\square G_2) \\leq \\tau(G_1)\\tau(G_2) [\\frac{2m_1}{n_1-1} + \\frac{2m_2}{n_2-1}]^{(n_1-1)(n_2-1)}.$$ We also characterize the graphs for which equality holds. As a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6340","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"}