{"paper":{"title":"Degree Majorization and Laplacian Eigenvalue Sums for Simplicial Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huan-Zhi Zhang, Yi-Min Song, Yi-Zheng Fan","submitted_at":"2026-07-23T04:20:41Z","abstract_excerpt":"Let $K$ be an $r$-dimensional simplicial complex. We prove that the spectrum of its $(r - 1)$-dimensional up-Laplacian is majorized by the conjugate degree sequence of its $(r - 1)$-dimensional faces: \\[ {\\mathbf{\\lambda}}_{r-1}(K) \\preccurlyeq {\\mathbf d}_{r-1}^\\top(K). \\] We also establish a Brouwer-type inequality: for every integer $\\ell \\geq 1$, \\[ \\sum_{i = 1}^{\\ell}\\lambda_{r-1,i}(K) \\leq \\frac{r + 1}{2}f_r(K) +\n\\frac{f_{r - 2}(K)}{r} \\binom{\\ell + 1}{2}, \\] where $\\lambda_{r-1,i}(K)$ denotes the $i$-th largest eigenvalue in the spectrum ${\\mathbf{\\lambda}}_{r-1}(K)$, and $f_t(K)$ denot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20910","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.20910/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}