# The average size of matchings in graphs

## TL;DR

This paper investigates the average size of matchings in graphs, characterizes extremal cases, and establishes inequalities relating matchings to other graph invariants, advancing understanding of graph matching properties.

## Contribution

It provides a characterization of extremal graphs for average matching size and introduces inequalities linking matchings with matching energy and counts.

## Key findings

- Characterization of extremal graphs for average matching size
- Inequalities between average matching size and matching number
- Relations between matching energy and matching polynomial zeros

## Abstract

In this paper, we consider the average size of independent edge sets, also called matchings, in a graph. We characterize the extremal graphs for the average size of matchings in general graphs and trees. In addition, we obtain inequalities between the average size of matchings and the number of matchings as well as the matching energy, which is defined as the sum of the absolute values of the zeros of the matching polynomial.