We have carried out an intensive study of the AGN heating-ICM cooling network by comparing various cluster parameters of the HIFLUGCS sample to the integrated radio luminosity of the central AGN, L_R, defined as the total synchrotron power between 10
MHz and 15 GHz. We adopt the central cooling time, t_cool, as the diagnostic to ascertain cooling properties of the clusters and classify clusters with t_cool < 1 Gyr as strong cooling core (SCC) clusters, with 1 Gyr < t_cool <7.7 Gyr as weak cooling core (WCC) clusters and with t_cool > 7.7 Gyr as non-cooling core (NCC) clusters. We find 48 out of 64 clusters (75%) contain cluster center radio sources (CCRS) cospatial with or within 50 h^{-1}_{71} kpc of the X-ray peak emission. Further, we find that the probability of finding a CCRS increases from 45% to 67% to 100% for NCC, WCC and SCC clusters, respectively, suggesting an AGN-feedback machinery in SCC clusters which regulates the cooling in the central regions. We find L_R in SCC clusters depends strongly on the cluster scale such that more massive clusters harbor more powerful radio AGN. The same trend is observed between L_R and the classical mass deposition rate, MDR, albeit much stronger, in SCC and partly also in WCC clusters. We also perform correlations of the 2MASS K-band luminosity of the brightest cluster galaxy, L_BCG, with L_R and cluster parameters. We invoke the relation between L_BCG and the black hole mass, M_BH, and find a surprisingly tight correlation between M_BH and L_R for SCC clusters. We find also an excellent correlation of L_BCG with M500 and L_X for the entire sample; however, SCC clusters show a tighter trend in both the cases. We discuss the plausible reasons behind these scaling relations in the context of cooling flows and AGN feedback. [Abridged]
