  Diffuse radio emission in galaxy clusters is known to be related to cluster
mass and cluster dynamical state. We collect the observed fluxes of radio
halos, relics, and mini-halos for a sample of galaxy clusters from the
literature, and calculate their radio powers. We then obtain the values of
cluster mass or mass proxies from previous observations, and also obtain the
various dynamical parameters of these galaxy clusters from optical and X-ray
data. The radio powers of relics, halos, and mini-halos are correlated with the
cluster masses or mass proxies, as found by previous authors, with the
correlations concerning giant radio halos being, in general, the strongest
ones. We found that the inclusion of dynamical parameters as the third
dimension can significantly reduce the data scatter for the scaling relations,
especially for radio halos. We therefore conclude that the substructures in
X-ray images of galaxy clusters and the irregular distributions of optical
brightness of member galaxies can be used to quantitatively characterize the
shock waves and turbulence in the intracluster medium responsible for
re-accelerating particles to generate the observed diffuse radio emission. The
power of radio halos and relics is correlated with cluster mass proxies and
dynamical parameters in the form of a fundamental plane.
