Abstract
Let \( G \) be a simple connected graph with vertex set \( V(G) \) and edge set \( E(G) \). A formal definition of a vertex-degree-based topological index (VDB topological index) of \( G \) is \begin{align*} &\mathcal{TI}_f(G)=\sum_{uv\in E}f(d_G(u),d_G(v)),
\end{align*} where \( f(x,y)>0 \) is a symmetric real function with \( x\ge 1 \) and \( y\ge 1 \), and \( d_G(u) \) is the degree of vertex \( u \) in \( G \).
In this paper, we give some conditions related to the function \( f(x,y) \), and show that if a VDB topological index satisfies these conditions, then the extremal graphs must be almost regular. From this conclusion, we obtained the
minimum/maximum values of such VDB topological indices among \( c \)-cyclic graphs, and characterize the extremal \( c \)-cyclic graphs that achieve the minimum/maximum values. As an application, we show that there are many VDB topological
indices that satisfy the conditions given in this paper. These VDB topological indices include the second Zagreb index, reciprocal Randić index, first hyper-Zagreb index, first Gourava index, second Gourava index, product-connectivity
Gourava index, exponential reciprocal sum-connectivity index, exponential inverse degree index, first Zagreb index, forgotten index, inverse degree index, Sombor index, reduced Sombor index, third Sombor index, fourth Sombor index, and
so on.
