Abstract
The question of finding extremal structures with respect to various graph indices has received a lot of attention. Among these indices, a large number are defined on vertex degrees. We consider a typical generalization of the vertex-degree based indices of a graph \( G \) defined by
\[
I_{f}(G)=\sum_{uv\in E(G)} f(d(u), d(v)),
\]
where \( f(x,y) \) is symmetric bivariate function. We define a property concerning \( f(x,y) \) and show that if \( f(x,y) \) admits this property and \( G \) has a given matching number, then \( I_{f}(G) \) is upper bounded by a graph with certain structure. Further, we show that the above property is admitted by a large number of degree-based indices. This means that the extremal structures of the graphs that have given matching number and attain the maximum values of these indices are the same.
