  We present the first study on the mass functions of Jenkins et al (J01) and
an estimate of their corresponding largest virialized dark halos in the
Universe for a variety of dark-energy cosmological models with a running
spectral index. Compared with the PL-CDM model, the RSI-CDM model can raise the
mass abundance of dark halos for small mass halos at lower redshifts, but it is
not apparent on scales of massive mass halos. Particularly, this discrepancy
increases largely with the decrease of redshift, and the RSI-CDM model can
suppress the mass abundance on any scale of halo masses at higher redshift. As
for the largest mass of virialized halos, the spatially flat $\Lambda$CDM
models give more massive mass of virialized objects than other models for both
of PL-CDM and RSI-CDM power spectral indexs, and the RSI-CDM model can enhance
the mass of largest virialized halos for all of models considered in this
paper. So we probably distinguish the PL-CDM and RSI-CDM models by the largest
virialized halos in the future survey of cluster of galaxies.
