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A certain signed adjacency matrix of the hypercube, which Hao Huang used lastyear 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 whichthis connection is a natural first example of the relationship between grouplabeled adjacency matrices with few eigenvalues, and combinatoriallyinteresting covering graphs. In particular, we define a two-eigenvalue cover tobe a covering graph whose adjacency spectra differs (as a multiset) from thatof the graph it covers by exactly two eigenvalues. We show that walk regularityof a graph implies walk regularity of any abelian two-eigenvalue cover. We alsogive a spectral characterization for when a cyclic two-eigenvalue cover of astrongly-regular graph is distance regular.

Graph covers with two new eigenvalues