{"paper":{"title":"Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jernej Azarija, Riste \\v{S}krekovski","submitted_at":"2012-10-23T19:34:26Z","abstract_excerpt":"Let $\\alpha(n)$ be the least number $k$ for which there exists a simple graph with $k$ vertices having precisely $n \\geq 3$ spanning trees. Similarly, define $\\beta(n)$ as the least number $k$ for which there exists a simple graph with $k$ edges having precisely $n \\geq 3$ spanning trees. As an $n$-cycle has exactly $n$ spanning trees, it follows that $\\alpha(n),\\beta(n) \\leq n$. In this paper, we show that $\\alpha(n) \\leq \\frac{n+4}{3}$ and $\\beta(n) \\leq \\frac{n+7}{3} $ if and only if $n \\notin {3,4,5,6,7,9,10,13,18,22}$, which is a subset of Euler's idoneal numbers. Moreover, if $n \\not \\eq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6335","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"}