Extremal degree-based topological indices for trees with given segment number

A degree-based topological index of a tree $T$ is termed as

$$T{I}_f(T)=\sum _{v_1{v}_2\in E(T)}f(d(v_1),d(v_2)),$$

in which $f(x,y)=f(y,x)$ denotes a real-valued function with $x,y\ge 1$. This paper mainly focuses on the extremal topological index problems for trees with a given segment number. We respectfully present the sufficient conditions for achieving the smallest and largest values of $T{I}_f$, as well as depict the associated extremal graphs. As an application, it is verified that there are eight types of degree-based indices that meet these sufficient conditions, including the recently proposed Euler Sombor index and diminished Sombor index.