{"paper":{"title":"Loose Laplacian spectra of random hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Xing Peng","submitted_at":"2011-09-15T19:12:18Z","abstract_excerpt":"Let $H=(V,E)$ be an $r$-uniform hypergraph with the vertex set $V$ and the edge set $E$. For $1\\leq s \\leq r/2$, we define a weighted graph $G^{(s)}$ on the vertex set ${V\\choose s}$ as follows. Every pair of $s$-sets $I$ and $J$ is associated with a weight $w(I,J)$, which is the number of edges in $H$ passing through $I$ and $J$ if $I\\cap J=\\emptyset$, and 0 if $I\\cap J\\not=\\emptyset$. The $s$-th Laplacian $\\L^{(s)}$ of $H$ is defined to be the normalized Laplacian of $G^{(s)}$. The eigenvalues of $\\mathcal L^{(s)}$ are listed as $\\lambda^{(s)}_0, \\lambda^{(s)}_1,..., \\lambda^{(s)}_{{n\\choose"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3433","kind":"arxiv","version":3},"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"}