On the number of $r$-matchings in a Tree

TL;DR

This paper investigates the maximum number of $r$-matchings, including induced matchings, in trees of fixed size, establishing that paths maximize induced matchings among all trees of the same order.

Contribution

It provides bounds on the number of $r$-matchings in trees and proves that paths have the greatest number of induced matchings among all $n$-vertex trees.

Findings

Paths have the maximum number of induced matchings among all trees.

The paper estimates the maximum number of $r$-matchings in trees.

It extends understanding of matching structures in graph theory.

Abstract

An $r$-matching in a graph $G$ is a collection of edges in $G$ such that the distance between any two edges is at least $r$. A $2$-matching is also called an induced matching. In this paper, we estimate the maximum number of $r$-matchings in a tree of fixed order. We also prove that the $n$-vertex path has the maximum number of induced matchings among all $n$-vertex trees.