Signless Laplacian spectral radius and fractional matchings in graphs

Abstract

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e\in Γ(v)}f(e)\leq 1$ for each $v\in V(G)$, where $Γ(v)$ is the set of edges incident to $v$. The {\it fractional matching number} of $G$, written $α'_{*}(G)$, 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.