Abstract
A topological index reflects the physical, chemical and structural properties of a molecule, and its study has an important role in molecular topology, chemical graph theory and mathematical chemistry. It is a natural problem to characterize non-isomorphic
graphs with the same topological index value. By introducing a relation on trees with respect to edge division vectors, denoted by \(\langle\mathcal{T}_n, \preceq \rangle\), in this paper we give some results for the relation order in
\(\langle\mathcal{T}_n, \preceq \rangle\). It allows us to compare the size of the topological index value without relying on the specific forms of them, and naturally we can determine which trees have the same topological index value.
Based on these results we characterize some classes of trees that are uniquely determined by their edge division vectors. Moreover we construct infinite classes of non-isomorphic trees with the same topological index value, particularly
such trees of order no more than 10 are completely determined.