Ratio of the number of $\mathbf{1}$-nearly independent vertex subsets and the Merrifield-Simmons index

The number ${\sigma }_k(G)$ of induced subgraphs with size $k$ of a graph $G$ was introduced recently as the number of $k$-nearly independent vertex subsets of $G$. Results highlighting similarity and difference in the behaviours of ${\sigma }_1$ and ${\sigma }_0$, have been reported. In this paper, we provide more comparison tools, by studying the ratio $\frac{{\sigma }_1(G)}{{\sigma }_0(G)}$. We establish sharp lower and upper bounds for this ratio over various classes of graphs, including connected graphs, trees, and forests.