Spectral Radius and Fractional Perfect Matchings in Graphs

For an n-vertex graph G, a fractional matching of G is a function f giving each edge a real number in [0, 1] such that Sigma(e is an element of Gamma (v)) f (e) <= 1 for each vertex v is an element of V(G), where Gamma(v) is the set of edges incident to v. A fractional perfect matching is a fractional matching f with Sigma(e is an element of E(G)) f (e) = n/2. In this paper, we establish tight lower bounds on the size and the spectral radius of G to guarantee that G has a fractional perfect matching, respectively. In addition, we investigate the relationship between fractional perfect matching and P->= 2-factor, and give some sufficient conditions for a graph to have a P->= 2-factor.