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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 $S_{r}$. 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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 $S_{r}$. 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