A general VDB topological index of $G$ is defined as

$${\mathcal{T}\mathcal{I}}_f(G)=\sum _{v_i{v}_j\in E(G)}f(d_i,d_j),$$

where $f(d_i,d_j)$ is a symmetric function, and $d_i$ represents the degree of $v_i\in V(G)$. This paper aims to address the maximum chemical tree problem for general vertex-degree-based topological indices with given number of pendant vertices via a unified method. Sufficient conditions for general VDB topological indices to take their maximum value are presented, and as an application, we show that there are six VDB topological indices, including the reciprocal sum-connectivity index, the Sombor index, and the Euler Sombor index, etc., that satisfy these conditions.

The bond incident degree index of $G$ is defined as

$$BID(G)=\sum _{u_1{u}_2\in E(G)}f(d(u_1),d(u_2)),$$

where $f(y,x)=f(x,y)$ is a real-valued function. In this paper, using graph transformation methods, we respectively established the maximum bond incident degree indices of trees and unicyclic graphs with a fixed diameter. As an application of the sufficient conditions, we verified that six bond incident degree indices satisfy such conditions, among which are the newly introduced Euler Sombor index and the computationally complex general Sombor index.

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.