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Hi everyone, we know that the lifted(covering) graph inherits every eigenvalue of the base graph. Friedman called eigenvalues of base graph the "old eigenvalues" of its lift. My question is, what can we say about the "new eigenvalues" of a $k$-lift? For $k=2$(double cover), the new eigenvalues are known to be the eigenvalues of the signed adjacency matrix. But for $k\geq 3$, no similar nice result exists. So far, for the sake of constructing Ramanujan graphs, the study of spectra of lifts has focused on randaom lifts(where the perfect matching between two fibers are selected at random) of regular graphs. People use probability method to show the existence of lift with "new eigenvalues" within a certain range. Though the adjancency matrix of covering graph is quite special: a block matrix whose blocks are either zero matrices or permutation matrices. There are few results(for some very special lift like universal cover) on the "new eigenvalues". Is it too much to ask for a non-trival bound for some of the "new eigenvalues" or an nice decompositon of the spectra of lifts like what we have for 2-lift? Maybe it is just hopeless since the structrue of an arbitary lift is too random and may have weak connection with the base graph.

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