Let graph \( G \) be an ordered pair \( (V(G),E(G)) \) consisting of a set \( V(G) \) of vertices and a set \( E(G) \) of edges. The formal definition of a vertex-degree-based topological index (VDB topological index) of \( G \) is \( TI_f(G)=\sum\limits_{v_{i}v_{j}\in E(G)}f(d(v_{i}),d(v_{j})) \), where \( f(x,y)> 0 \) is a symmetric real function with \( x \ge 1 \) and \( y\ge 1 \), and \( d(v_{i}) \) denotes the degree of vertex \( v_{i} \) in \( G \). When \( f(x, y) = (\frac{x+y-2}{xy})^\alpha \), where \( \alpha \) is an arbitrary non-zero real number, this VDB topological index is called the generalized atom-bond connectivity index (or \( ABC_\alpha \) index for short) of \( G \), which was introduced as a topological index by Furtula et al.

In this paper, we present several conditions on the function \( f(x, y) \) and prove that if a VDB topological index satisfies these conditions, then its extremal graphs are almost regular. Based on this conclusion, we derive the maximum VDB indices of trees and unicyclic graphs with perfect matchings, and characterize the corresponding extremal graphs. As an application, we verify that the ABC\( _\alpha \) index for \( 0<\alpha \le1 \) satisfies the conditions given in this paper, present the maximum ABC\( _\alpha \) index of trees and unicyclic graphs with perfect matchings, and characterize the corresponding extremal graphs. Our work thereby extends several previously known results.