A Note on the Extremal Graphs with Respect to Vertex-Degree-Based Topological Indices for c-Cyclic Graphs

A general VDB topological index of G is defined aswhere f (x, y) is a real symmetric function for x 1 and y 1.Recently, Liu et al. (2024) presented a uniform method for solving the extremal problem with general VDB topological indices for ccyclic 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 T I 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.