There are different ways to distinguish the graphs, for example, using the complexity index and also using the vector complexity index. Is there a study on the efficiency of the two methods?
It is based on the user requirement. In general, the vector complexity index is more applicable. See e.g., https://www.inet.ox.ac.uk/files/main_feb4.pdf
In fact, I am now looking for a comparative study between the two aforementioned methods to find out the efficiency of the two methods in distinguishing the graphs and determining the most efficient method, although just as you kindly said, it may depend on the reality of the applied study.