Abstract
Let \( G \) be a simple connected graph. For a vertex-degree-based topological index \( TI_f(G)=\sum\limits_{uv\in E(G)}f(d_u,d_v) \), where \( f(x,y) \) is a pertinently chosen symmetric real function, the topological index \( RTI_f(G)=\sum\limits_{uv\in
E(G)}\frac{1}{f(d_u,d_v)} \) is called the reciprocal index of \( TI_f \). In this paper, for the first Zagreb index (\( f(x,y)=x+y \)), the second Zagreb index (\( f(x,y)=xy \)), and the forgotten index (\( f(x,y)=x^2+y^2 \)), we prove
that the star \( S_n \) and the path \( P_n \) achieve the maximum and minimum values of \( TI_f+RTI_f \) among all trees of order \( n \), respectively. In addition, we show that the same conclusion holds for some other vertex-degree-based
topological indices.