{"paper":{"title":"The maximum $p$-Spectral Radius of Hypergraphs with $m$ Edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu","submitted_at":"2018-03-23T04:22:35Z","abstract_excerpt":"For $r\\geq 2$ and $p\\geq 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $$\\rho_p(H)=\\max_{{\\bf x}\\in \\mathbb{R}^n: \\|{\\bf x}\\|_p=1}r \\cdot \\!\\!\\!\\! \\sum_{\\{i_1,i_2,\\ldots, i_r\\}\\in E(H)} x_{i_1}x_{i_2}\\cdots x_{i_r},$$ where the maximum is taken over all ${\\bf x\\in \\mathbb{R}^n}$ with the $p$-norm equals 1. In this paper, we proved for any integer $r\\geq 2$, and any real $p\\geq 1$, and any $r$-uniform hypergraph $H$ with $m={s\\choose r}$ edges (for some real $s\\geq r-1$), we have $$\\lambda_p(H)\\leq \\frac{rm}{s^{r/p}}.$$ The equality holds if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.08653","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"}