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

TL;DR

This paper investigates the ratio of nearly independent vertex subsets in graphs, establishing bounds for this ratio across different graph classes to deepen understanding of their combinatorial properties.

Contribution

It introduces bounds for the ratio of $\sigma_1(G)$ to $\sigma_0(G)$, providing new comparison tools for analyzing graph subset structures.

Findings

Established sharp bounds for the ratio in various graph classes

Compared behaviors of $\sigma_1$ and $\sigma_0$ in different graphs

Provided new insights into the structure of nearly independent vertex subsets

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.