In this paper, we examine the uniqueness (discrimination power) of a newly proposed graph invariant based on the matrix [Formula: see text] defined by Randić et al. In order to do so, we use exhaustively generated graphs instead of special graph classes such as trees only. Using these graph classes allow us to generalize the findings towards complex networks as they usually do not possess any structural constraints. We obtain that the uniqueness of this newly proposed graph invariant is approximately as low as the uniqueness of the Balaban [Formula: see text] index on exhaustively generated (general) graphs.