{"paper":{"title":"Triangle Families with Large Edge Up-Laplacian Spectral Gap","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mutasim Mim","submitted_at":"2026-05-26T17:18:11Z","abstract_excerpt":"Let $\\mathcal{T}$ be a finite nonempty set of $3$-element subsets of a totally ordered set $V$. We view $\\mathcal{T}$ as the set of triangles in the support graph. Let $\\delta_{1,\\mathcal{T}}$ be the signed edge-triangle incidence matrix, and $\\lambda(\\mathcal{T})$ the spectral gap of $\\delta_{1,\\mathcal{T}}^T\\delta_{1,\\mathcal{T}}.$\n  Our main results show that large $\\lambda(\\mathcal{T})$ forces strong overlap and a large minimum degree in the support graph. In particular, every support edge lies in at least $\\lceil \\lambda(\\mathcal{T})\\rceil-2$ triangles in $\\mathcal{T}$ and hence the graph"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27307","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/2605.27307/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"}