{"paper":{"title":"A Spectral Lower Bound on Chromatic Numbers using $p$-Energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Clive Elphick, Quanyu Tang, Shengtong Zhang","submitted_at":"2025-04-02T01:59:22Z","abstract_excerpt":"Let $A_G $ be the adjacency matrix of a simple graph $ G $, and let $ \\chi(G) $, $ \\chi_f(G) $, $ \\chi_q(G) $, $ \\xi(G) $ and $ \\xi_f(G) $ denote its chromatic number, fractional chromatic number, quantum chromatic number, orthogonal rank and projective rank, respectively. For $ p \\geq 0 $, we define the positive and negative $ p $-energies of $ G $ by $$ \\mathcal{E}_p^+(G) = \\sum_{\\lambda_i > 0} \\lambda_i^p, \\quad \\mathcal{E}_p^-(G) = \\sum_{\\lambda_i < 0} |\\lambda_i|^p, $$ where $ \\lambda_1 \\geq \\cdots \\geq \\lambda_n $ are the eigenvalues of $A_G $. We prove that for all $ p \\geq 0 $, $$ \\chi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.01295","kind":"arxiv","version":5},"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/2504.01295/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"}