Graph covers with two new eigenvalues

A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve sensitivity conjecture, is closely related unique, 4-cycle free, 2-fold cover hypercube. We develop a framework in this connection natural first example relationship between group labeled matrices with few eigenvalues, and combinatorially interesting covering graphs. In particular, we define two-eigenvalue be graph whose spectra differs (as multiset) from that it covers by exactly two eigenvalues. show walk regularity implies any abelian cover. also give spectral characterization for when cyclic strongly-regular distance regular.