The number of 1-nearly independent edge subsets

TL;DR

This paper investigates the number of 1-nearly independent edge subsets in graphs, characterizing extremal trees and proposing a conjecture for the second-largest case.

Contribution

It characterizes the extremal trees with smallest and largest $Z_1$ and proposes a conjecture for the second-largest $Z_1$ among $n$-vertex trees.

Findings

Identified the two $n$-vertex trees with smallest $Z_1$.

Identified the $n$-vertex tree with largest $Z_1$.

Proposed a conjecture for the second-largest $Z_1$.

Abstract

Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. A subset $S$ of $E(G)$ is called a $k$-nearly independent edge subsets if there are exactly $k$ pairs of elements of $S$ that share a common end. $Z_k(G)$ is the number of such subsets. This paper studies $Z_1$. Various properties of $Z_1$ are discussed. We characterise the two $n$-vertex trees with smallest $Z_1$, as well as the one with largest value. A conjecture on the $n$-vertex tree with second-largest $Z_1$ is proposed.