{"paper":{"title":"The critical exponent: a novel graph invariant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CO","authors_text":"Apoorva Khare, Bala Rajaratnam, Dominique Guillot","submitted_at":"2018-02-20T06:09:35Z","abstract_excerpt":"A surprising result of FitzGerald and Horn (1977) shows that $A^{\\circ \\alpha} := (a_{ij}^\\alpha)$ is positive semidefinite (p.s.d.) for every entrywise nonnegative $n \\times n$ p.s.d. matrix $A = (a_{ij})$ if and only if $\\alpha$ is a positive integer or $\\alpha \\geq n-2$. Given a graph $G$, we consider the refined problem of characterizing the set $\\mathcal{H}_G$ of entrywise powers preserving positivity for matrices with a zero pattern encoded by $G$. Using algebraic and combinatorial methods, we study how the geometry of $G$ influences the set $\\mathcal{H}_G$. Our treatment provides new an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.06976","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"}