Graph covers with two new eigenvalues

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