Signless Laplacian spectral radius and fractional matchings in graphs

TL;DR

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

Contribution

It establishes new bounds and relations connecting the fractional matching number with the signless Laplacian spectral radius, including conditions for fractional perfect matchings.

Findings

Spectral bounds for fractional matching number

Conditions for fractional perfect matchings based on spectral radius

Relations between graph spectra and fractional matchings

Abstract

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e\in \Gamma(v)}f(e)\leq 1$ for each $v\in V(G)$, where $\Gamma(v)$ is the set of edges incident to $v$. The {\it fractional matching number} of $G$, written $\alpha'_{*}(G)$, is the maximum of $\sum_{e\in E(G)}f(e)$ over all fractional matchings $f$. In this paper, we propose the relations between the fractional matching number and the signless Laplacian spectral radius of a graph. As applications, we also give sufficient spectral conditions for existence of a fractional perfect matching in a graph in terms of the signless Laplacian spectral radius of the graph and its complement.