We implement the spectral renormalization group on different deterministic
non-spatial networks without translational invariance. We calculate the
thermodynamic critical exponents for the Gaussian model on the Cayley tree and
the diamond lattice, and find that they are functions of the spectral
dimension, d~. The results are shown to be consistent with those from
exact summation and finite size scaling approaches. At d~=2, the lower
critical dimension for the Ising universality class, the Gaussian fixed point
is stable with respect to a ψ4 perturbation up to second order. However,
on generalized diamond lattices, non-Gaussian fixed points arise for
2<d~<4.Comment: 16 pages, 14 figures, 5 tables. The paper has been extended to
include a ψ4 interactions and higher spectral dimension