Signless Laplacian spectral radius and fractional matchings in graphs

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that ${\sum }_{e\in \mathrm{\Gamma }(v)}f(e)\le 1$ for each $v\in V(G)$,where$\Gamma(v)$isthesetofedgesincidentto$v$. The {\it fractional matching number} of $G$, written ${\alpha }_{\ast }^{\mathrm{\prime }}(G)$, 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.