  We use the ROSAT all sky survey X-ray cluster catalogs and the optical SDSS
DR7 galaxy and group catalogs to cross-identify X-ray clusters with their
optical counterparts, resulting in a sample of 201 X-ray clusters in the sky
coverage of SDSS DR7. We investigate various correlations between the optical
and X-ray properties of these X-ray clusters, and find that the following
optical properties are correlated with the X-ray luminosity: the central galaxy
luminosity, the central galaxy mass, the characteristic group luminosity
($\propto \Lx^{0.43}$), the group stellar mass ($\propto \Lx^{0.46}$), with
typical 1-$\sigma$ scatter of $\sim 0.67$ in $\log \Lx$. Using the observed
number distribution of X-ray clusters, we obtain an unbiased scaling relation
between the X-ray luminosity, the central galaxy stellar mass and the
characteristic satellite stellar mass as ${\log L_X} = -0.26 + 2.90 [\log
(M_{\ast, c} + 0.26 M_{\rm sat}) -12.0]$ (and in terms of luminosities, as
${\log L_X} = -0.15 + 2.38 [\log (L_{c} + 0.72 L_{\rm sat}) -12.0]$). We find
that the systematic difference between different halo mass estimations, e.g.,
using the ranking of characteristic group stellar mass or using the X-ray
luminosity scaling relation can be used to constrain cosmology. Comparing the
properties of groups of similar stellar mass (or optical luminosities) and
redshift that are X-ray luminous or under-luminous, we find that X-ray luminous
groups have more faint satellite galaxies and higher red fraction in their
satellites. The cross-identified X-ray clusters together with their optical
properties are provided in Appendix B.
