  Magnetization, specific heat, and electron spin resonance (ESR) measurements
are carried out to clarify the low-energy excitations for the S = 1/2
polyhedral clusters {Mo$_{72}$V$_{30}$} and {W$_{72}$V$_{30}$}. The clusters
provide unique model systems of Kagom\'e network on a quasi-sphere. The linear
field variation of magnetization at low temperatures indicates that the ground
state is singlet for both clusters. The temperature and the magnetic field
dependence of specific heat shows a distinct difference between two clusters
with differing structural symmetries. In {W$_{72}$V$_{30}$} with I$_h$
symmetry, the existence of the several tens of low-energy singlet excited
states below the lowest triplet excited state is revealed. The specific heat of
the slightly distorted {Mo$_{72}$V$_{30}$} with D$_{5h}$ symmetry leads to a
drastic decrease of the singlet contribution, which is consistent with the
partial lifting of the frustration and the decrease of degenerated low-energy
states. The singlet excitation existence is confirmed further by the
temperature dependence of the ESR spectra. Comprehensive experimental studies
have demonstrated unique low-energy excitations of the spherical Kagom\'e
networks and their sensitivity to the cluster symmetry.
