  A middle-cube is an induced subgraph consisting of nodes at the middle two
layers of a hypercube. The middle-cubes are related to the well-known Revolving
Door (Middle Levels) conjecture. We study the middle-cube graph by completely
characterizing its spectrum. Specifically, we first present a simple proof of
its spectrum utilizing the fact that the graph is related to Johnson graphs
which are distance-regular graphs and whose eigenvalues can be computed using
the association schemes. We then give a second proof from a pure graph theory
point of view without using its distance regular property and the technique of
association schemes.
