The COVID-19 pandemic, caused by the SARS-CoV-2 virus, has triggered myriad efforts to dissect and understand the structure and dynamics of this complex pathogen. The Spike glycoprotein of SARS-CoV-2 has received special attention as it is the means by which the virus enters the human host cells. The N-terminal domain (NTD) is one of the targeted regions of the Spike protein for therapeutics and neutralizing antibodies against COVID-19. Though its function is not well-understood, the NTD is reported to acquire mutations and deletions that can accelerate the evolutionary adaptation of the virus driving antibody escape. Cellular processes are known to be regulated by complex interactions at the molecular level, which can be characterized by means of a graph representation facilitating the identification of key residues and critical communication pathways within the molecular complex. From extensive all-atom molecular dynamics simulations of the entire Spike for the wild-type and the dominant variant, we derive a weighted graph representation of the protein in two dominant conformations of the receptor-binding-domain; all-down and one-up. We implement graph theory techniques to characterize the relevance of specific residues at facilitating roles of communication and control, while uncovering key implications for fitness and adaptation. We find that many of the reported high-frequency mutations tend to occur away from the critical residues highlighted by our graph theory analysis, implying that these mutations tend to avoid targeting residues that are most critical for protein allosteric communication. We propose that these critical residues could be candidate targets for novel antibody therapeutics. In addition, our analysis provides quantitative insights of the critical role of the NTD and furin cleavage site and their wide-reaching influence over the protein at large. Many of our conclusions are supported by empirical evidence while others point the way towards crucial simulation-guided experiments.