Abstract
A general vertex-degree-based (VDB) topological index of a graph \(G\) is defined as
\[
\mathcal{T}_f=\mathcal{T}_f(G)=\sum_{uv\in E(G)}f(d_G(u),d_G(v)),
\]
where \(f(x,y)>0\) is a symmetric real function with \(x\ge 1\) and \(y\ge 1\). Let \({\cal CT}_n\) be the set of all chemical trees of order \(n\), and let \(\hat{T}_f= \max \{ \mathcal{T}_f(T)\mid T\in {{\cal CT}_n}\}\).
A chemical tree \(T\in {\cal CT}_n\) is an \(n\)-optimal \(\mathcal{T}_f\) chemical tree if \(\mathcal{T}_f(T)=\hat{T}_f\).
One important topic in chemical graph theory is the extremal value problem of VDB topological indices over \({\cal CT}_n\). In this work, we get the following results.
(1) We propose six conditions (C1)-(C6) for the symmetric real function \(f(x,y)\). For a VDB topological index \(\mathcal{T}_f\) satisfied the conditions (C1)-(C6), we obtained the necessary and sufficient conditions for \(T\in {{\cal CT}_n}\) to be an \(n\)-optimal \(\mathcal{T}_f\) chemical tree.
(2) For twenty-five VDB topological indices (as shown in Table 4.1 of Section 4), the \(n\)-optimal \(\mathcal{T}_f\) chemical trees are characterized, and the maximum \(\mathcal{T}_f\) values are determined, too.
