Topological indices, also known as molecular descriptors, are an indispensable tool for quantitative structure property/activity relationships models, which play a significant role in drug design processes. Randic presented the properties that a topological index must possess in chemical graph theory to be considered a good molecular descriptor. Among the prominent properties is the fact that a topological index should change gradually with a gradual change in structures. An effective and widely accepted method for measuring this feature was first proposed by Furtula et al., based on graph edit distance. This method was later improved by Rakic and Furtula drawing on the $z$-score associated with the Tanimoto index. This paper performs a comparative analysis of the structure sensitivity and abruptness values of graph energy types, based on fourteen different topological indices, using these two methods.