{"paper":{"title":"Spectral Radius of $\\{0, 1\\}$-Tensor with Prescribed Number of Ones","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Linyuan Lu, Shuliang Bai","submitted_at":"2018-01-09T03:45:22Z","abstract_excerpt":"For any $r$-order $\\{0, 1\\}$-tensor $A$ with $e$ ones, we prove that the spectral radius of $A$ is at most $e^{\\frac{r-1}{r}}$ with the equality holds if and only if $e={k^r}$ for some integer $k$ and all ones forms a principal sub-tensor ${\\bf 1}_{k\\times \\cdots \\times k}$. We also prove a stability result for general tensor $A$ with $e$ ones where $e=k^r+l$ with relatively small $l$. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all $r$-order $\\{0, 1\\}$-tensor $A$ with $k^r+l$ ones, for $-r-1\\leq l \\leq r$, and $k$ sufficientl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02784","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"}