  Many popular graph metrics encode average properties of individual network
elements. Complementing these conventional graph metrics, the eigenvalue
spectrum of the normalized Laplacian describes a network's structure directly
at a systems level, without referring to individual nodes or connections. In
this paper, we study the spectrum and their applications of normalized
Laplacian matrices of hype-cubes, a special kind of Cayley graphs. We determine
explicitly all the eigenvalues and their corresponding multiplicities by a
recursive method. By using the relation between normalized Laplacian spectrum
and eigentime identity, we derive the explicit formula to the eigentime
identity for random walks on the hype-cubes and show that it grows linearly
with the network size. Moreover, we compute the number of spanning trees of the
hype-cubes.
