The equilibrium properties and the Rouse-Zimm dynamics of polymer molecules with any architecture at temperatures T less than or equal to Theta, are treated using a bead-and-spring coarse-grained description. The collapsed globule model is adopted, whereby essentially all atoms are at the same mean-square distance (S-2) from the center of mass; accordingly, at a given temperature the interatomic free energy is a single-valued function of (S-2) and the self-consistent free-energy minimization is greatly simplified, in the Gaussian approximation. We prove that the connectivity matrix B and the bond-vector product matrix M=[(I-i.I-j)] possess the same eigenvectors; these are the normal modes of the chain conformation. Furthermore, we show that (S-2)=N(at)(-1)Sigma(k)l(2) alpha(k)(2)/lambda(k), where N-at is the total number of atoms, lambda(k) is the general nonzero eigenvalue of B, and l(2) alpha(k)(2) is the corresponding eigenvalue of M-the expansion ratio of the normal mode. Finally, we prove that in the free-draining limit the normal mode relaxation times are proportional to l(2) alpha(k)(2)/lambda(k). Defining as alpha(s)=root(S-2)/(S-2)(ph) as the overall strain ratio with respect to the phantom state, the plots of alpha(s) vs the reduced temperature tau=(T-Theta)/T less than or equal to 0 indicate that polymers with more compact architectures display a prompter contraction for small tau’s, although tending to larger alpha(s)’s at strong undercoolings, where the average density (alpha N-at.(S-2)(-3/2)) is about the same for all architectures. Concerning the dynamical behavior, at sufficiently large tau’s the longest relaxation times reach a typical plateau, as already found for the linear chain.