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

Abstract

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.