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 valued 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.

郝煌去年在解决超敏感猜想时使用了某种超立方体的有符号邻接矩阵，该矩阵与超立方体的独特，无4周期，2折覆盖率密切相关。我们开发了一个框架，其中这种连接是特征值很少的群值邻接矩阵与组合有趣的覆盖图之间关系的自然第一个例子。特别是，我们定义了一个两个特征值Cover，它是一个覆盖图，其邻接光谱与它所覆盖的图的邻接光谱恰好有两个特征值不同（作为多集）。我们证明了图的行走规律性暗示了任何阿贝尔二特征值覆盖率的行走规律性。当强规则图的循环二特征值覆盖范围是距离规则的时，我们还给出了频谱特征。