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 ∑ _e∈Γ (v)f(e)≤ 1 for each vertex v∈ V(G) , where Γ (v) is the set of edges incident to v . A fractional perfect matching is a fractional matching f with ∑ _e∈ 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.