  We derive the determinant of the Laplacian for the Hanoi networks and use it
to determine their number of spanning trees (or graph complexity)
asymptotically. While spanning trees generally proliferate with increasing
average degree, the results show that modifications within the basic patterns
of design of these hierarchical networks can lead to significant variations in
their complexity. To this end, we develop renormalization group methods to
obtain recursion equations from which many spectral properties can be obtained.
This provides the basis for future applications to explore the physics of
several dynamic processes.
