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arXiv · 2407.09067

An improved lower bound on the number of $1$-nearly independent vertex subsets

Zekhaya B. Shozi
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Abstract

Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. For $k\ge 0$ an integer, a subset $I_k$ of $V(G)$ is called a $k$-nearly independent vertex subset of $G$ if $I_k$ induces a subgraph of size $k$ in $G$. The number of such subsets in $G$ is denoted by $σ_k(G)$. In this paper we continue the study of $σ_1$. In particular, we prove the lower bound on $σ_1$ for a connected graph that contains a cycle and also characterise the two extremal graphs. This improves the result obtained in [E. O. D. Andriantiana and Z. B. Shozi. The number of 1-nearly independent vertex subsets. \textit{Quaestiones Mathematicae}, accepted].

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Zekhaya B. Shozi. 2024-07-12. An improved lower bound on the number of $1$-nearly independent vertex subsets. https://arxiv.org/abs/2407.09067

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