  Topological indices are important bridge between graph theory and chemical
applications. In this paper, we will provide some sufficient conditions for a
graph to be $k$-extendable graphs, $n$-factor-critical graphs and
$(n,k,d)$-graphs in terms of a significant type of Topological index: the
Zeroth-order General Randi\'c Index ($\alpha\ge-1$ and $\alpha\neq 0$).
