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arXiv · 2604.11813

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

Audace A. V. Dossou-Olory·Eric O. D. Andriantiana·Valisoa R. M. Rakotonarivo·Zekhaya B. Shozi
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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.

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Audace A. V. Dossou-Olory, Eric O. D. Andriantiana, Valisoa R. M. Rakotonarivo, Zekhaya B. Shozi. 2026-04-10. Ratio of the number of $\mathbf{1}$-nearly independent vertex subsets and the Merrifield-Simmons index. https://arxiv.org/abs/2604.11813

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