  We study a family of graphs related to the $n$-cube. The middle cube graph of
parameter $k$ is the subgraph of $Q_{2k-1}$ induced by the set of vertices
whose binary representation has either $k-1$ or $k$ number of ones. The middle
cube graphs can be obtained from the well-known odd graphs by doubling their
vertex set. Here we study some of the properties of the middle cube graphs in
the light of the theory of distance-regular graphs. In particular, we
completely determine their spectra (eigenvalues and their multiplicities, and
associated eigenvectors).
