A relation on trees and the topological indices based on subgraph

TL;DR

This paper introduces a new relation on trees based on edge division vectors to compare topological indices, characterizes trees uniquely determined by these indices, and constructs classes of non-isomorphic trees sharing the same index.

Contribution

It presents a novel relation on trees that helps compare topological indices without specific calculations and characterizes trees uniquely identified by their edge division vectors.

Findings

The relation order allows comparison of topological index values.

Some classes of trees are uniquely determined by their edge division vectors.

Infinite classes of non-isomorphic trees with the same topological index are constructed.

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 and 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.