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We have carried out an intensive study of the AGN heating-ICM cooling networkby comparing various cluster parameters of the HIFLUGCS sample to theintegrated radio luminosity of the central AGN, L_R, defined as the totalsynchrotron 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 andclassify 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 witht_cool > 7.7 Gyr as non-cooling core (NCC) clusters. We find 48 out of 64clusters (75%) contain cluster center radio sources (CCRS) cospatial with orwithin 50 h^{-1}_{71} kpc of the X-ray peak emission. Further, we find that theprobability of finding a CCRS increases from 45% to 67% to 100% for NCC, WCCand SCC clusters, respectively, suggesting an AGN-feedback machinery in SCCclusters which regulates the cooling in the central regions.  We find L_R in SCC clusters depends strongly on the cluster scale such thatmore massive clusters harbor more powerful radio AGN. The same trend isobserved between L_R and the classical mass deposition rate, MDR, albeit muchstronger, in SCC and partly also in WCC clusters. We also perform correlationsof the 2MASS K-band luminosity of the brightest cluster galaxy, L_BCG, with L_Rand cluster parameters. We invoke the relation between L_BCG and the black holemass, M_BH, and find a surprisingly tight correlation between M_BH and L_R forSCC clusters. We find also an excellent correlation of L_BCG with M500 and L_Xfor the entire sample; however, SCC clusters show a tighter trend in both thecases. We discuss the plausible reasons behind these scaling relations in thecontext of cooling flows and AGN feedback. [Abridged]

AGN heating and ICM cooling in the HIFLUGCS sample of galaxy clusters