The unfolding of a protein can be described as a transition from a predominantly rigid, folded structure to an ensemble of denatured states. During unfolding, the hydrogen bonds and salt bridges break, destabilizing the secondary and tertiary structure. Our previous work shows that the network of covalent bonds, salt bridges, hydrogen bonds, and hydrophobic interactions forms constraints that define which regions of the native protein are flexible or rigid (structurally stable). Here, we test the hypothesis that information about the folding pathway is encoded in the energetic hierarchy of non-covalent interactions in the native-state structure. The incremental thermal denaturation of protein structures is simulated by diluting the network of salt bridges and hydrogen bonds, breaking them one by one, from weakest to strongest. The structurally stable and flexible regions are identified at each step, providing information about the evolution of flexible regions during denaturation. The folding core, or center of structure formation during folding, is predicted as the region formed by two or more secondary structures having the greatest stability against denaturation. For 10 proteins with different architectures, we show that the predicted folding cores from this flexibility/stability analysis are in good agreement with those identified by native-state hydrogen-deuterium exchange experiments.