Trees maximizing the number of almost-perfect matchings

TL;DR

This paper characterizes extremal trees with maximum almost-perfect matchings and minimal maximal matchings, and applies these findings to optimize the weighted Hosoya index based on vertex degrees.

Contribution

It provides a complete characterization of trees that maximize or minimize specific types of matchings, advancing understanding of extremal properties in graph theory.

Findings

Identified trees with maximum almost-perfect matchings.

Determined trees with minimum maximal matchings.

Applied results to optimize the weighted Hosoya index.

Abstract

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.