{"paper":{"title":"Upper Bounds for the Largest Laplacian Eigenvalue of Simplicial Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huan-Zhi Zhang, Yi-Zheng Fan","submitted_at":"2026-06-19T08:54:01Z","abstract_excerpt":"Let $K$ be a finite $r$-dimensional simplicial complex with vertex set $V$ of size $n$. We study the largest eigenvalue of the combinatorial $(r-1)$-up Laplacian $L^{\\operatorname{up}}_{r-1}(K)$. It is known that \\[ \\lambda_{\\max}\\bigl(L^{\\operatorname{up}}_{r-1}(K)\\bigr)\\le n. \\] We first give a homological equality criterion for this universal bound, namely, the equality holds if and only if the $r$-dimensional complement $K^c$ of $K$ has a nonzero reduced homology $\\widetilde H_{r-1}(K^c,\\mathbb{R})$. For $r=1$, this is the classical graph condition that the complement graph is disconnected"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21233","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/2606.21233/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"}