# Signless Laplacian spectral radius and fractional matchings in graphs

## TL;DR

This paper explores the relationship between the fractional matching number and the signless Laplacian spectral radius of graphs, providing spectral conditions for fractional perfect matchings.

## Contribution

It establishes new connections between spectral graph theory and fractional matchings, including sufficient conditions for fractional perfect matchings based on spectral properties.

## Key findings

- Derived bounds linking fractional matching number and spectral radius.
- Provided spectral criteria ensuring the existence of fractional perfect matchings.

## Abstract

A fractional matching of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ such that $\sum_{e\in\Gamma(v)}f(e)\leq1$ for each vertex $v\in V(G)$, where $\Gamma(v)$ is the set of edges incident to $v$. The fractional matching number of $G$, written $\alpha^{\prime}_*(G)$, is the maximum value of $\sum_{e\in E(G)}f(e)$ over all fractional matchings. In this paper, we investigate the relations between the fractional matching number and the signless Laplacian spectral radius of a graph. Moreover, we give some sufficient spectral conditions for the existence of a fractional perfect matching.