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

TL;DR

This paper establishes a new lower bound for the number of 1-nearly independent vertex subsets in connected graphs with cycles, and characterizes the extremal graphs achieving this bound.

Contribution

It improves previous bounds on -nearly independent subsets and characterizes extremal graphs for this property.

Findings

Proved a lower bound on  for connected graphs with cycles.

Characterized the extremal graphs achieving the bound.

Enhanced understanding of nearly independent vertex subsets in graph theory.

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 $\sigma_k(G)$. In this paper we continue the study of $\sigma_1$. In particular, we prove the lower bound on $\sigma_1$ 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].