Topological indices are important bridge between graph theory and chemical applications. The study of graph matching expandability has been an influential topic in recent research on graph structure. In this paper, we provide some sufficient conditions for a graph to be $k$-extendable graphs, $n$-factor-critical graphs and $(n,k,d)$-graphs and graphs with prefect matching in terms of a significant type of Topological index: the Zeroth-order General Randić Index ($α\ge-1$ and $α\neq 0$).