  The low-energy structure and $\beta$ decay properties of neutron-rich even-
and odd-mass Pd and Rh nuclei are studied using a mapping framework based on
the nuclear density functional theory and the particle-boson coupling scheme.
Constrained Hartree-Fock-Bogoliubov calculations using the Gogny-D1M energy
density functional are performed to obtain microscopic inputs to determine the
interacting-boson Hamiltonian employed to describe the even-even core Pd
nuclei. The mean-field calculations also provide single-particle energies for
the odd systems, which are used to determine essential ingredients of the
particle-boson interactions for the odd-nucleon systems, and of the
Gamow-Teller and Fermi transition operators. The potential energy surfaces
obtained for even-even Pd isotopes as well as the spectroscopic properties for
the even- and odd-mass systems suggest a transition from prolate deformed to
$\gamma$-unstable and to nearly-spherical shapes. The predicted $\beta$ decay
$\log{ft}$ values are shown to be sensitive to the details of the wave
functions for the parent and daughter nuclei, and therefore serve as a
stringent test of the employed theoretical approach.
