{"paper":{"title":"More on the phi = beta Conjecture and Eigenvalues of Random Graph Lifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Doron Puder, Edward Lui","submitted_at":"2009-09-07T12:12:45Z","abstract_excerpt":"Let $G$ be a connected graph, and let $\\lambda_1$ and $\\rho$ denote the spectral radius of $G$ and the universal cover of $G$, respectively. In \\cite{Fri03}, Friedman has shown that almost every $n$-lift of $G$ has all of its new eigenvalues bounded by $O(\\lambda_1^{1/2}\\rho^{1/2})$. In \\cite{LP10}, Linial and Puder have improved this bound to $O(\\lambda_1^{1/3}\\rho^{2/3})$. Friedman had conjectured that this bound can actually be improved to $\\rho + o_n(1)$ (e.g., see \\cite{Fri03,HLW06}).\n  In \\cite{LP10}, Linial and Puder have formulated two new categorizations of formal words, namely $\\phi$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1231","kind":"arxiv","version":2},"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"}