{"paper":{"title":"Ramanujan Coverings of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Chris Hall, Doron Puder, William F. Sawin","submitted_at":"2015-06-08T01:54:10Z","abstract_excerpt":"Let $G$ be a finite connected graph, and let $\\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\\rho=2\\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan $r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus, Spielman and Srivastava (2013).\n  Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by elements of the symmetric group $S_{r}$. We generali"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02335","kind":"arxiv","version":3},"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"}