We implement the spectral renormalization group on different deterministic nonspatial 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) over tilde. The results are shown to be consistent with those from exact summation and finite-size scaling approaches. At (d) over tilde = 2, the lower critical dimension for the Ising universality class, the Gaussian fixed point is stable with respect to a psi(4) perturbation up to second order. However, on generalized diamond lattices, non-Gaussian fixed points arise for 2 < <(d)over tilde> < 4.