Let G n be the set of graphs with n vertices and H ⊆ G n .For each H ∈ H, let m (H) = {m i,j (H)}, where m i,j (H) is the number of edges in H that join a vertex of degree i with a vertex of degree j.A vertex-degree-based (VDB, for short) topological index φ is discriminating over H if non-isomorphic graphs in H have different values of φ.We say that φ is weakly discriminating over H if the following weaker condition is satisfied for every H, H ′ ∈ H:Let CT n be the set of chemical trees with n vertices.In this paper we show that many of the well-known VDB topological indices are not weakly discriminating over CT n .However, the recently introduced Sombor index is weakly discriminating over CT n .Also, we give conditions under which a VDB topological index φ is weakly discriminating over an arbitrary class H ⊆ G n .