A fractional matching of a graph G is a function f:E(G)→[0,1]such that for each vertex v,∑e∈ΓG(v)f(e)≤1,where ΓG(v)is the set of edges incident with v.The fractional matching number μf(G)of G is the maximum value of ∑e∈E(G)f(e)overallfractional matchings f.Liu and Liu(2002)obtained the relationship between the fractional matching number and the matching number of a graph G which is that μf(G)=μ(G)+nc(G)/2 In this paper,we characterize the saturated graph with a given fractional matching number by using this formula,where the saturated graph is a graph G such that μf(G+uv)＞μf(G)for any two nonadjacent vertices u and v of G.Among n-vertex graphs with given fractional matching number,we characterize the extremal graph that has the minimum distance signless Laplacian spectral radius,give an upper bound and a lower bound of signless Laplacian spectral radius,and characterize the extremal graph that has the maximum signless Laplacian spectral radius.
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