Signless Laplacian spectral radius and fractional matchings in graphs

Abstract

A fractional matching of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ such that ${\sum }_{e\in \Gamma (v)}f(e)\le 1$ 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 ${\alpha }_{\ast }^{\prime }(G)$, 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.