  By assigning a probability measure via the spectrum of the normalized
Laplacian to each graph and using L^p Wasserstein distances between probability
measures, we define the corresponding spectral distances d_p on the set of all
graphs. This approach can even be extended to measuring the distances between
infinite graphs. We prove that the diameter of the set of graphs, as a
pseudo-metric space equipped with d_1, is one. We further study the behavior of
d_1 when the size of graphs tends to infinity by interlacing inequalities
aiming at exploring large real networks. A monotonic relation between d_1 and
the evolutionary distance of biological networks is observed in simulations.
