A general VDB topological index of \( G \) is defined as \[ \mathcal{TI}_f=\mathcal{TI}_f(G)=\sum_{uv\in E(G)} f(d(u),d(v)), \] where \( f(x,y) \) is a real symmetric function for \( x\geq 1 \) and \( y\geq 1 \). Recently, Liu et al. (2024) presented a uniform method for solving the extremal problem with general VDB topological indices for \( c \)-cyclic graphs, which was later extended by Gao (2025) and Ali et al. (2025). In this note, we further investigate this problem. A new mathematical formula for \( \mathcal{TI}_f \) was obtained, which provided sufficient conditions for \( G \) to take its minimum value. As an application, we show that there are sixteen VDB topological indices that satisfy these conditions. In addition, we ordered six VDB topological indices for bicyclic and tricyclic graphs.