  Two new classes of networks are introduced that resemble small-world
properties. These networks are recursively constructed but retain a fixed,
regular degree. They consist of a one-dimensional lattice backbone overlayed by
a hierarchical sequence of long-distance links. Both types of networks, one
3-regular and the other 4-regular, lead to distinct behaviors, as revealed by
renormalization group studies. The 3-regular networks are planar, have a
diameter growing as \sqrt{N} with the system size N, and lead to
super-diffusion with an exact, anomalous exponent d_w=1.3057581..., but
possesses only a trivial fixed point T_c=0 for the Ising ferromagnet. In turn,
the 4-regular networks are non-planar, have a diameter growing as
~2^[\sqrt(\log_2 N^2)], exhibit "ballistic" diffusion (d_w=1), and a
non-trivial ferromagnetic transition, T_c>0. It suggest that the 3-regular
networks are still quite "geometric", while the 4-regular networks qualify as
true small-world networks with mean-field properties. As an example of an
application we discuss synchronization of processors on these networks.
