The number of $1$-nearly independent vertex subsets

TL;DR

This paper investigates the count of subsets in a graph that contain exactly one adjacent pair, providing formulas and bounds, and identifying extremal trees for this measure.

Contribution

It introduces the concept of 1-nearly independent vertex subsets, derives recursive and explicit formulas, and establishes bounds and extremal graph structures for their count.

Findings

Recursive formulas for $\sigma_1$ are derived.

Tight bounds on $\sigma_1$ for graphs of order $n$ are proved.

The star $K_{1,n-1}$ minimizes $\sigma_1$ among trees.

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A subset $I$ of $V(G)$ is an independent vertex subset if no two vertices in $I$ are adjacent in $G$. We study the number, $\sigma_1(G)$, of all subsets of $v(G)$ that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of $\sigma_1$ are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on $\sigma_1$ for graphs of order $n$. We deduce as a corollary that the star $K_{1,n-1}$ (the tree with degree sequence $(n-1,1,\dots,1)$) is the $n$-vertex tree with smallest $\sigma_1$, while it is well known that $K_{1,n-1}$ is the $n$-vertex tree with largest number of independent subsets.